Analysis - Sequences and Series | AI tutor The No.1 Homework Finishing Free App

'Find the sequence p_n from the above result.'

'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.'

'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).'

'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'

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

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

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

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

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

'Example Number Practical 2 Recurrence Relation'

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

'How to find the general term'

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

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

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

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

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

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

'Find the infinite series related to area.'

"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 \"

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

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

'Principle of Credit Creation'

'Consider the following infinite series:'

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

'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)'

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

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

'Practice proving the following infinite series diverges.'

'Number of shapes and combinations.'

'Find the sum of the following series.'

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

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

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

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

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

'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'

'Find the following sum.'

'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.'

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

'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.'

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

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

'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.'

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

'Comprehensive Exercise'

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

'Example 126 Infinite Series and Limits (2)'

'Example 11 | Convergence and Divergence of Infinite Series'

'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\\]'

'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 }'

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

'Prove the following integral inequalities.'

'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'

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

'Prove the 22nd equation of long book'

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

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

'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 $\\lim _{n \\rightarrow \\infty} S_{n}$.'

'Infinite series'

'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〕'

'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 $\\sum_{n=1}^{\\infty}(k a_{n}+l b_{n})=k S+l T$.'

'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)'

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

'(2) $\\sum_{n=2}^{\\infty} \\frac{1}{n^{2}-1}$'

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

'Limit of infinite geometric sequence { r^n }'

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

'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) \\).'

'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})'

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

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

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

'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.'

'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.'

'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.'

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

'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 \\).'

'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'

'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 \'

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

'Find the limit of the following sequences.'

'Find the sum of the following infinite series.'

'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'

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

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

'Investigate the limits of the following sequences.'

'Prove that the following infinite series diverges.'

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

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

'Find the sum of the following infinite series.'

'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.'

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

'Infinite geometric series'

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

'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.'

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

'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)} \\)'

'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\'

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