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## Functions and Analysis

### Analysis - Sequences and Series

#### Q.02

'When the k-th term of a sequence a_k can be expressed as a_k=f(k+1)-f(k), find the formula for \\sum_{k=1}^{n} a_k using the equation (*) below and explain the reasoning behind it.'

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'By deriving only (2) and (4), from the fact that the general term of the differenced sequence of the sequence {pn} is (-1/2)^(n+1), find the general term pn, or by deriving only (1) and (3), solve the recurrence relation between adjacent terms (3).'

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'Practice: Find the general term of the sequence {an} determined by the following conditions. (1) a1=1, an+1=3an+2n-1 (2) a1=-30,9an+1=an+43n'

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'Please explain how to solve a system of simultaneous recurrence equations.'

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'Find the sum S_{n} of an arithmetic sequence {a_{n}} with the first term a, common difference d, and number of terms n.'

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'Therefore, the sequence {a_{n}+20} is a geometric sequence with initial term 2 and common ratio 5/4.'

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'Find the general term of the sequence $\\left\\{a_{n}\\right\\}$ determined by the following conditions.'

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'Find the general term of a sequence using the formula for the sum'

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'Given the sum Sn of the series from the first term to the nth term, provide the formula to find the general term.'

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'How to find the general term from a recurrence relation'

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'Example question number Practice 1 Arithmetic sequence, geometric sequence'

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'Explain arithmetic sequence and general term, and find the general term based on an example.'

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'Find the sum of the first n terms of the given sequence.'

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'Find the limit value of the sequence {an} determined by the following conditions. a1=0, a2=1, an+2=1/4(an+1+3an)'

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'Find the limit of the sequence \ \\left\\{a_{n}\\right\\} \ determined by the following conditions.'

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"Find the sum of the following infinite series.\n(1) Let \ \\left\\{a_{n}\\right\\} \ be a geometric progression with first term 2 and common ratio 2. Find \ \\sum_{n=1}^{\\infty} \\frac{1}{a_{n} a_{n+1}} \ (similar to Aichi Institute of Technology)\n(2) Let \ \\pi \ be the constant representing the ratio of a circle's circumference to its diameter. Evaluate \ 1+\\frac{2}{\\pi}+\\frac{3}{\\pi^{2}}+\\frac{4}{\\pi^{3}}+\\cdots \\cdots+\\frac{n+1}{\\pi^{n}}+\\cdots \\dots \\nYou may use the fact that \\( \\lim _{n \\rightarrow \\infty} n x^{n}=0(|x|<1) \\) if needed.\n(Similar to Keio University) \ \\rightarrow 33,35 \"

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'Find the limit of the sequence determined by the following conditions:'

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'Find the sum of the series $S_{n} = \\sum_{k=1}^{n} k\\left(\\frac{1}{4}\\right)^{k-1}$.'

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'Using the result from the example above, prove that the infinite series $\\sum_{n=1}^{\\infty} \\frac{1}{\\sqrt{n}}$ diverges.'

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'For convergent series $\\sum_{n=1}^{\\infty} a_{n}, \\sum_{n=1}^{\\infty} b_{n}$, where $\\sum_{n=1}^{\\infty} a_{n}=S, \\sum_{n=1}^{\\infty} b_{n}=T$, then the series $\\sum_{n=1}^{\\infty}(k a_{n}+l b_{n})$ will also converge and $\\sum_{n=1}^{\\infty}(k a_{n}+l b_{n})=k S+l T$ (where $k, l$ are constants)'

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'Please solve a problem related to the sum of an infinite series.'

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Join our Discord#### Q.28

'Please verify the convergence of the sequence {1/n^k} when k>0.'

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'Practice proving the following infinite series diverges.'

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'State the conditions for a sequence to be an arithmetic progression.'

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Join our Discord#### Q.33

'Prove by mathematical induction that the general term formula guessed in (2) (1) is correct.'

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'Find the sum of the series: \\(\\sum_{k=1}^n(k^2+3k+1)\\)'

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'Find the sum from the first term to the nth term of the sequence.'

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'Find the general term of the following sequence: $c_{n+1}=-2c_{n}-18$'

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'Find the 5th term of the sequence defined by the following conditions: (1) a₁=1, aₙ₊₁=3aₙ-1, (2) a₁=0, aₙ₊₁=-3aₙ+2n'

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Join our Discord#### Q.39

'Prove that if the series $\\sum_{n=1}^{\\infty} a_{n}, \\sum_{n=1}^{\\infty} b_{n}$ converge, where $\\sum_{n=1}^{\\infty} a_{n}=S, \\sum_{n=1}^{\\infty} b_{n}=T$, then the series $\\sum_{n=1}^{\\infty}\\left(k a_{n}+l b_{n}\\right)$ also converges and find its sum. Here, $k, l$ are constants.'

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'Show the convergence condition of an infinite geometric series.'

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'Explain and prove the convergence and divergence of an infinite geometric series. Determine the conditions under which the infinite series created from the infinite geometric sequence {ar^n-1} with initial term a and common ratio r \ \\sum_{n=1}^{\\infty} \overline^{n-1}=a+\overline+\overline^{2}+\\cdots \\cdots+\overline^{n-1}+\\cdots \\cdots \ converges, find its sum, and indicate the conditions under which it diverges.'

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Join our Discord#### Q.42

'Find the sum of the series \\( \\sum_{n=1}^{\\infty}\\left(\\frac{1}{3}\\right)^{n} \\sin \\frac{n \\pi}{2} \\).'

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'Investigate the convergence and divergence of an infinite geometric series, and if it converges, find its sum.'

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'For the infinite series $x+\\frac{x}{1+x}+\\frac{x}{(1+x)^{2}}+\\cdots \\cdots+\\frac{x}{(1+x)^{n-1}}+\\cdots \\cdots$,\n(1) Determine the range of values of $x$ for which this infinite series converges.\n(2) Let $f(x)$ be the sum of this infinite series when $x$ is in the range determined in (1). Draw the graph of the function $y=f(x)$ and investigate its continuity.'

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'Prove that the infinite series Σ(1 / n) diverges.'

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'Investigate the limit of the infinite geometric series {r^n}.'

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'Example 11 | Convergence and Divergence of Infinite Series'

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'Investigate the convergence or divergence of the following infinite series, and if it converges, find its sum. \n\\[\n\\left(2-\\frac{1}{2}\\right)+\\left(\\frac{2}{3}+\\frac{1}{2^{2}}\\right)+\\left(\\frac{2}{3^{2}}-\\frac{1}{2^{3}}\\right)+\\cdots \\cdots+\\left(\\frac{2}{3^{n-1}}+\\frac{(-1)^{n}}{2^{n}}\\right)+\\cdots \\cdots\n\\]'

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'Investigate the convergence and divergence of the following sequences: (a) { -n^{3} + 1 } (b) { -\\frac{1}{n^{3}} + 2 } (c) { \\frac{3}{n+2} } (d) { \\frac{(-2)^{n}}{3} - 1 }'

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'Convergence and Divergence of Infinite Series and Limit of Terms'

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'Exercise 14 |II| → Booklet p.343\n(1) Assuming the sequence {xn} converges, and its limit value is α\nlim_{n→∞} xn = lim_{n→∞} xn+1 = α\nTherefore, as n → ∞, xn+1 = √(a + xn)\nα = √(a + α)\nSquaring and rearranging both sides gives α² - α - a = 0\nSince α > 0, α = (1 + √(1 + 4a)) / 2'

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'Find the sum of the infinite series ∑ from n=0 to infinity of (1/2)^n cos(nπ/6).'

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'Using the intermediate value theorem to solve infinite series sum and definite integral'

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'Explain the limit of an infinite geometric series and indicate under what conditions it converges.'

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'For the sequence defined by $b_{n}=(-1)^{n-1} \\log _{2} \\frac{n+2}{n}(n=1,2,3, \\ldots \\ldots)$, let $S_{n}=b_{1}+b_{2}+\\cdots \\cdots+b_{n}$. Find $\\lim _{n \\rightarrow \\infty} S_{n}$.'

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Join our Discord#### Q.61

'Prove that the sequence of points Pn(x_{n}, y_{n}) that satisfies the recurrence relation and limits (4) for a system of linear equations P1(1, 1), x_{n+1}=\x0crac{1}{4} x_{n}+\x0crac{4}{5} y_{n}, y_{n+1}=\x0crac{3}{4} x_{n}+\x0crac{1}{5} y_{n}(n=1,2, ...) on the plane. Prove that the points P1, P2, ... approach a certain fixed point infinitely close.〔Similar to Shinshu University〕'

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Join our Discord#### Q.62

'Prove that for two convergent series $\\sum_{n=1}^{\\infty} a_{n}, \\sum_{n=1}^{\\infty} b_{n}$ converging to $S, T$ respectively, for constants $k, l$, the series $\\sum_{n=1}^{\\infty}(k a_{n}+l b_{n})$ also converges and $\\sum_{n=1}^{\\infty}(k a_{n}+l b_{n})=k S+l T$.'

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Join our Discord#### Q.63

'Find the definite integral \ \\int_{0}^{1} x^{2} d x \. First, divide the interval \ [0, 1] \ into \ n \ equal parts, and let the area of each rectangle with blue shadow be \\( \\left(\\frac{1}{n}\\right) \\cdot\\left(\\frac{k}{n}\\right)^{2} \\ (k=0,1, \\cdots, n-1) \\), then the sum is\n\\[S_{n} = \\sum_{k=0}^{n-1} \\frac{1}{n} \\cdot\\left(\\frac{k}{n}\\right)^{2} = \\frac{1}{n^{3}} \\sum_{k=1}^{n-1} k^{2} = \\frac{1}{6}\\left(1 - \\frac{1}{n}\\right)\\left(2 - \\frac{1}{n}\\right)'

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Join our Discord#### Q.64

'Find the limit value of the sequence \ \\left\\{a_{n}\\right\\} \ determined by the following conditions.'

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Join our Discord#### Q.66

'Find the limit value of the sequence {an} determined by the following conditions.'

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'Investigate the convergence and divergence of the following infinite series, and find the sum if it converges.'

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Join our Discord#### Q.69

'Investigate the convergence or divergence of the infinite series \ \\sum_{n=1}^{\\infty} n x^{n-1} \, and find the sum if it converges. You can use \\( \\lim _{n \\rightarrow \\infty} n x^{n}=0(|x|<1) \\).'

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Join our Discord#### Q.70

'Find the volume V of the solid obtained by rotating the region enclosed by the following two curves around the x-axis by 1 revolution. (1) y=x^{2}-2, y=2x^{2}-3 (2) y=√3 x^{2}, y=√(4-x^{2})'

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'Prove that the series 1+\\frac{1}{2}+\\frac{1}{3}+\\cdots diverges.'

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'Find the sum of this infinite series: 1-1/3+1/5-1/7+⋯⋯'

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'Practice investigating the convergence and divergence of the following infinite series, and find the sum if it converges.'

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Join our Discord#### Q.74

'A cube with a mark on one face is placed on a horizontal plane. One of the four edges of the base of the cube is randomly chosen with equal probability, and the cube is tilted sideways around this edge n times. Let the probability of the marked face facing upward be aₙ, and facing downward be bₙ. Assume that the initially marked face is facing upward. (1) Find a₂. (2) Express aₙ₊₁ in terms of aₙ. (3) Find limₙ→∞ aₙ.'

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Join our Discord#### Q.75

'Using the result from (2), find the sum of the infinite series Σ_(n=1)^∞ n/2^n. You may use the fact that lim (n→∞)(n/2^n)=0.'

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Join our Discord#### Q.76

'For the infinite series $x+\\frac{x}{1+x}+\\frac{x}{(1+x)^{2}}+\\cdots \\cdots+\\frac{x}{(1+x)^{n-1}}+\\cdots \\cdots$, find (1) the range of values for $x$ for which this infinite series converges. (2) Let $f(x)$ denote the sum of this infinite series when $x$ is in the range found in (1). Draw the graph of the function $y=f(x)$ and investigate its continuity.'

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'Find the limit value of the sequence $\\{a_{n} \\}$ determined by the following conditions.'

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'Find the sum of the infinite series \\(\\left(1-\\frac{1}{2}\\right)+\\left(\\frac{1}{3}-\\frac{1}{2^{2}}\\right)+\\left(\\frac{1}{3^{2}}-\\frac{1}{2^{3}}\\right)+\\cdots \\cdots \\).'

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'Chapter 3\nDifferential Calculus - 105\nWhen n ≥ 2\n\\[\egin{aligned}b_{n} & =b_{1}+\\sum_{k=1}^{n-1} 6 k=0+6 \\cdot \\frac{1}{2}(n-1) n \\& =3 n(n-1)\\end{aligned}\\]\nThis holds even when n=1.\n\\[\\text{Therefore,} \\quad b_{n}=3 n(n-1)\\]\n\\( \\sum_{k=1}^{n} k=\\frac{1}{2} n(n+1) \\)\nChapter 3\nEX'

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Join our Discord#### Q.80

'Find the sum of the following infinite series.\n(1) \\(\\left(1+\\frac{2}{3}\\right)+\\left(\\frac{1}{3}+\\frac{2^{2}}{3^{2}}\\right)+\\left(\\frac{1}{3^{2}}+\\frac{2^{3}}{3^{3}}\\right)+\\cdots \\cdots \\)\n(2) \\\frac{3^{2}-2}{4}+\\frac{3^{3}-2^{2}}{4^{2}}+\\frac{3^{4}-2^{3}}{4^{3}}+\\cdots \\cdots \'

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Join our Discord#### Q.84

'Prove that the following infinite series diverges. (1) 1+\\frac{2}{3}+\\frac{3}{5}+\\frac{4}{7}+\\cdots \\cdots (2) \\sin \\frac{\\pi}{2}+\\sin \\frac{3}{2} \\pi+\\sin \\frac{5}{2} \\pi+\\cdots \\cdots'

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'Find the range of real numbers for which the infinite series converges.'

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'Investigate the convergence and divergence of the following infinite series, and find the sum if it converges.'

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'Investigate the limits of the following sequences.'

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'Prove that the following infinite series diverges.'

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'Find the limit of the sequence determined by the following conditions.'

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'Find the limit of the sequence {an} determined by the following conditions.'

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Join our Discord#### Q.92

'Find the length of the following curves \ L \. (1) \ \\left\\{\egin{\overlineray}{l}x=e^{t} \\cos t \\y=e^{t} \\sin t\\end{\overlineray}\\right(0 \\leqq t \\leqq \\frac{\\pi}{2}\\\right.'

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Join our Discord#### Q.93

'Find the range of real numbers x that make the following sequences converge. Also, find the limit value at that time.'

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'Find the sum of the infinite series ∑_{n=0}^{∞}(1/2)^{n} cos (n π / 6).'

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Join our Discord#### Q.96

'Using the partial sum formula, find the convergence condition and sum of the infinite geometric series a+\overline+\overline^{2}+\overline^{3}+\\cdots+\overline^{n-1}+\\cdots.'

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Join our Discord#### Q.97

'(1) Infinite series \ \\sum_{n=1}^{\\infty} a_{n} \ converges \ \\Longrightarrow \\lim _{n \\rightarrow \\infty} a_{n}=0 \'

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'Find the sum of the following infinite series: (1) \\( \\sum_{n=2}^{\\infty} \\frac{\\log_{10}(1+\\frac{1}{n})}{\\log_{10}n \\log_{10}(n+1)} \\)'

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Join our Discord#### Q.99

'Find the limit of the sequence determined by the following conditions: \ \\left\\{a_{n}\\right\\} \ .\a_{1}=1, \\quad a_{n+1}=\\frac{2}{3} a_{n}+1\'

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'In learning mathematics, what are the important things to develop problem-solving skills? What is needed besides memorizing basic knowledge?'