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Results tagged with sequences-and-series
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user 503397
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.
1
vote
1
answer
390
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How to show a non positive series diverges? [closed]
I know only two convergence tests for non positive series Dirchlet's and Abel's. Both can only confirm the convergence.
Is there a test that can show if a non positive series diverges ? I expected it …
- 1,641
0
votes
3
answers
110
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Zeroes of $f_n(x)=(x^2+1)^{\frac32}+nx-2$ make a monotone sequence
$f_n(x)=(x^2+1)^{\frac32}+nx-2$
1) Prove that for every $n \in N $ there exist $x_n \in [0,2]$ such
that $f(x_n)=0$.
2) Show that $f_{n+1}(x_n)>0 $ and show that sequence {$x_n $}$_{n\i …
- 1,641
7
votes
1
answer
140
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Is $\lim\limits_{n\to\infty}\Bigl(\frac{2+\sin n}{3}\Bigr)^n =0 $; $n \in \mathbb{N}$?
$$\lim_{n\to\infty}\Bigl(\frac{2+\sin n}{3}\Bigr)^n $$ where $n$ is a natural number.
The problem is if the numerator is less than 3. I think it is because $\sin n$ is never $1$ otherwise $\pi$ woul …
- 1,641
0
votes
Solving $\lim_{n\to \infty}\frac{(1)(2)(3)\cdots (n)}{(n+1)(n+2)(n+3)\cdots(2n)}$
$$(n+1)(n+2)(n+3)\cdots2n\ge n^n$$
$$\frac{1}{(n+1)(n+2)(n+3)\cdots2n}\le \frac{1}{n^n}$$
$$0\le\lim_{n\to \infty}\frac{1\cdot2\cdot3\cdots n }{(n+1)(n+2)(n+3)\cdots 2n}\le \lim_{n\to \infty}\frac{n!} …
- 1,641
0
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Prove $\left[\ln(n + 1)\right]^p - \left[\ln(n)\right]^p \to 0$ as $n \to \infty$, $p \geq 1$.
$$\log(n+1)=\log(n(1+\frac1n))=\log n+\log(1+\frac1n)=\log n+\frac1n+o(\frac1n)$$
$$\log(n+1)^p=(\log n+\frac1n+o(\frac1n))^p =$$
$$\log^pn(1+\frac1{n\log n}+o(\frac1{n\log n}))^p=\log^pn(1+\frac p{n\ …
- 1,641
2
votes
2
answers
136
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$\sum_{n=1}^\infty\frac{(-1)^n\sin ^2n}{n}$ Is the following solution wrong ?; Does $\sum\fr...
$$\sum_{n=1}^\infty\frac{(-1)^n\sin ^2n}{n}$$
Solution from the lecture notes :
$$\frac{(-1)^n\sin ^2n}{n}=\frac{(-1)^n(1-\cos
> 2n)}{2n}=\frac{(-1)^n}{2n}-\frac{(-1)^n\cos 2n}{2n}$$
$\frac{(-1)^n}{2 …
- 1,641
1
vote
4
answers
97
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How do we know $\sum_{i=0}^n\frac{x^n}{n!} $ converges to $ e^x $ for all x? [duplicate]
$$\sum_{i=0}^n\frac{x^n}{n!} $$
I know the sum converges for all x but how do we know it converges to the expect value $e^x$.
This sum was derived as the Taylor series of $e^x$ around $0$. How do we …
- 1,641
2
votes
Determine for what values of x the series converges.8.4.7 Petrovic
assuming $a>b$ $$ \lim_{n\rightarrow \infty } \frac{x}{(a^n + b^n)^{1/n}}=\lim_{n\rightarrow \infty } \frac{x}{(a^n(1 + (b/a)^n))^{1/n}}= \lim_{n\rightarrow \infty } \frac{x}{a(1 + (b/a)^n)^{1/n}}= \f …
- 1,641