# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

43,933 questions
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### How to find the limit sum of a series [on hold]

Suppose $$S_n = \lim_{n\to\infty}\frac{\exp(i/2)}{\sum_{j=1}^{i}\exp(j/2)} \ \ \ \text{where} \ \ i = 1,\ldots,n$$ Programatically, $S_n\approx 0.49$ but would I show this by hand?
1answer
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### What is the next number in this series? 1, 2 ,3, 22, 4, 23, 5, 222, … [on hold]

I just know there has to be a pattern and that it has something to do with primes.
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### Let $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$. Prove that $x_{n+1} < x_{n}$

Let $$x_{n+1} = \frac{1}{2}(x_{n} + \frac{a}{x_{n}})$$ Prove that $x_{n+1} < x_{n}$ for $a \geq 0$. Hint: Let the initial guess satisfy $x_{1} > \sqrt{a}$ I am stuck at how to begin this. I ...
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### How to express this function as power series? [on hold]

How to express this function as power series $\frac{x}{(2+x^2)^2}$
2answers
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### Determine the series whether convergence or divergence with using ratio rest. [on hold]

This is the problem: $$\sum_{n=0}^\infty 3^n\sin((\frac{1}{4})^n)$$ I can't prove the convergence of this series, how can we solve it?
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1answer
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### Couple of questions on Hurwitz theorem

Hurwitz theorem as stated in Hahn and Epstein's Classical Complex Analysis is as follows: I've got couple of questions I'm stuck on. Is the converse of the Hurwitz theorem is true ? Provide an ...
3answers
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### How to prove the double sum of combinations is $3^n$

I have a double sum of combinations as follow $$S = \sum_{i=0}^{n}\sum_{k=i}^{n}{n \choose k}{k \choose i}.$$ I guessed and tested that $S = 3^n$, but I have no idea how to prove this. Any help is ...
1answer
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### Is the proportionality characteristic of this function being carried on?

$A=kx$ is a directly proportional function,where $A^2=B^2+C^2$.Does it necessarily mean $B$ and $C$ both vary directly with respect to $x$? If not, under what condition is this possible? Thank you in ...
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1answer
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### For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ [duplicate]

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ is convergent if and only if (1). $0<a<e$ (2). $0<a\leq e$ (3). $0<a<\frac{1}{e}$ (4). $0<a\leq \frac{1}{e}$ I tried ...
5answers
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### Sequence divergence test

Could I argue that $\left(1+\frac{1}{n}\right)^{n^2}$ = $e ^n$ therefore the sequence diverges? I am wondering if it is a legal move
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### Does the series $\sum\limits_{n=0}^{\infty}e^{-n}$ converge? [on hold]

I came upon this question , $\sum\limits_{n=0}^{\infty}e^{-n}$ converge ? where as no description of n is given , I tried to use something called power series , but here n is negative can I ...
2answers
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### what is $\lim_{x\to 0} f(x)$

Let $$f(x)=\sum_{n=1}^{\infty}{\sin(nx)\over n^2}$$ Then what is $\lim_{x\to 0} f(x)$ Now I know the series converges uniformly by $M-Test$ (Take $M_n=1/n^2$). What should be my next step. I am ...
3answers
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### How to solve this expression

$$\sum_{n=1}^\infty\frac1{(x-3)^{2n-1}}$$ How to change the given expression to a rational function
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### Find the sum of the function $\sum_{n=1}^\infty (\cos{x})^n$ on $(0, \pi)$ [on hold]

Find the sum of the function $\sum_{n=1}^\infty (\cos{x})^n$ on $(0, \pi)$ We have $-1\leq\cos{x}\leq 1$. So $(\cos{x})^n \to 0$ as $n\to \infty$ Please solve this problem. Please find the sum.
4answers
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### Limit of the converging infinite sequence $\frac{2e^{3n}-1}{e^{3n}+1}$.

What is the limit of $$\left\{\dfrac{2e^{3n}-1}{e^{3n}+1}\right\}_{n=1}^{\infty}$$ How do I deal with the 'n' when it is the power of e? What steps would I take and is there any site / video that ...
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### If $\sum a_n$ is convergent but not absolutely, then $\sum a_n^+$ diverges

Let $a_n \in \mathbb{R}$, such that $\sum_{n=1}^\infty|a_n|= \infty$ and $\sum_{n=1}^m a_n \to a$, as $m \to \infty$. Let $a_n^+=\max\{a_n,0\}.$ Show that $\sum_{n=1}^\infty a_n^+= \infty$. Approach: ...
2answers
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### Uniform convergence of series of function

The series of functions $f_n(x)= x^n/(1+x^n)$ is uniformly convergent on $[0,a]$ where $0<a<1$ and not uniformly on $[0,1)$. I have come across lots of problems like this , where the open ...
2answers
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