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For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
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Recurrent relation for a series: does this yields to divergence?
I have a series $\Sigma a_k X^k$ that satisfies: $$\frac{a_{k+2}}{a_k} = -\frac{l(l+1)-k(k+1)}{(k+1)(k+2)}$$ This yields to $$\frac{a_{k+2}}{a_k} {\sim}_{+\infty} 1$$ Assuming that $l$ never ... sequences-and-series convergence-divergence- 84
Alternating infinite series problem please solve [closed]
Please hint me how to solve this. $(1-\frac12+\frac13-\frac14+......) ^2 = 2[\frac12-\frac13(1+\frac12) +\frac14(1+\frac12+\frac13) -\frac15(1+\frac12+\frac13+\frac14) + ......]$ real-analysis calculus sequences-and-series- 1
Solve for the power series of $(1+x)^n$
I'm having a difficulty on solving this equation about power series. I am asked to solve: $(1+x)^n$ and I need to use this equation: $\sum_{n=0}^{\infty} ar^n= \frac{ar^2}{1-r}$ Lastly, I need to find ... calculus sequences-and-series power-series- 1
Prove: at most two circles are needed to be tangent to all the circle sequence
Construct a circle sequence $\{C_n\}$ (e.g., blue in the figure below) in 2D Cartesian coordiante system as: the $x$-coordiantes of centers of all the circle $C_n$ are $\frac{1}n$; all the circles $\{... sequences-and-series euclidean-geometry mobius-inversion- 684
Power series of $(1+x)^n$ [closed]
Solve for the $n$th term of the power series of $(1+x)^n$. where n=2 (include the notation) calculus sequences-and-series power-series- 1
Geometric sequences with trigonometry
For the following sum: $\cos(2\theta) + \cos^2(2\theta) + \cos^3(2\theta) ...+ \cos^N(2\theta)$ Why is the range $\{0 < \theta <\frac{\pi}{2}\}$ for there to be a sum? sequences-and-series trigonometry geometric-series- 1
Tricky convergence
I would appreciate any help on the following problem: Let p > 2 and let $(a_k)_{k \in \mathbb{N}}$ be a sequence of positive real numbers and let $a > 0$ be a real number. Assume further that $$... real-analysis sequences-and-series convergence-divergence- 157
Prove that $\lim_{n\to\infty} \sum_{k=1}^{n}$ $(\frac {k}{n^2})^{{k\over n^2}+1} = {1\over 2}$.
To simplify the question I imagined what if $k\over n^2$= $x$ then it will look like $x^{x+1}$.Else I couldn't think on how to proceed ahead. In one of my previously asked question viz. Prove that $\... real-analysis sequences-and-series- 235
Let $a_1$ be linearly independent to $a_2$ over $\mathbb{Q}.$ For $n\geq 3,$ let $ a_n = \vert a_{n-1} - a_{n-2} \vert.$ Does $\sum_n a_n\ $ converge?
Let $a_1$ be linearly independent to $a_2$ over the rational numbers. For $n\geq 3,\ $ let $ a_n = \vert a_{n-1} - a_{n-2} \vert.$ Does $\sum_n a_n\ $ converge? For example, let $a_1 = 1,\ a_2 = \ln 2=... sequences-and-series convergence-divergence euclidean-algorithm irrationality-measure- 12.5k
Sum to infinity. [closed]
Could someone please explain how this sum is calculated? $$\sum_{n=0}^{\infty} \frac{a^n e^{(-a)}}{n!} = 1$$ Thanks Edit: $$=e^{-a}\sum_{n=0}^{\infty} \frac{a^n }{n!}=e^{-a}e^{a}=1$$ sequences-and-series summation- 53
Determine n, for which $1! + 2! + ... + n!$ is a square of an integer [duplicate]
Let n be a natural number. Find all n's for which $1! + 2! + ... + n!$ is a square of an integer. Example: $n=3$, because $1! + 2! + 3! = 1 + 2 + 6 = 9 = 3^2$ (doesn't necessarily have to be same as ... sequences-and-series elementary-number-theory factorial- 1
Meaning of 'set of well-ordered sequences'
I'm trying to make sense of a construction of a module given in the following research paper: A New Construction of the Injective Hull, Fleischer, 1968. On the second page, a module $F$ is constructed,... sequences-and-series commutative-algebra modules well-orders injective-module- 2,550
$\sum_{n=1}^\infty \left(\frac{n^2-n+3}{n^2+3n-4}\right)^n\frac{x^{n-1}}{1+x^n} \left(x>0\right)$ Convergence/Divergence
Convergence/Divergence of \begin{equation*}\begin{aligned} \sum_{n=1}^\infty \left(\frac{n^2-n+3}{n^2+3n-4}\right)^n\frac{x^{n-1}}{1+x^n} \left(x>0\right) \end{aligned}\end{equation*} Let $a_n=\... calculus sequences-and-series- 2,023
Test for Convergence $\sum_{n=1}^\infty\frac{7^{3n}}{n!}$ and $\sum_{n=1}^\infty\sqrt{\ln\frac{n+5}{n+2}}$
Test the following series for convergence $$\sum_{n=1}^\infty\frac{7^{3n}}{n!}$$ and $$\sum_{n=1}^\infty\sqrt{\ln\frac{n+5}{n+2}}$$ I need this for studying purposes, I have an exam next week and I am ... sequences-and-series convergence-divergence power-series- 1
Lucas numbers relation to Φ
So, the Lucas numbers are 2,1,3,4,7,11... Let L(n) be nth lucas number Fibonacci numbers are 1,1,2,3,5,8,13,21... Φ^n=F(n+1)+F(n-1), F=Fibonacci number and n=nth So, if I say n=5, then Φ^n=F(6)+F(4)=... sequences-and-series fibonacci-numbers lucas-numbers- 1
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