0
votes
1answer
16 views

Absolute convergence.

Determine if absolutely convergent or not; Justify. $$\sum_{n=1}^\infty (-1)^n n^2 3^{1-n} x^n \text{ s.t }|x|<3$$ if we take the abs value of $(-1)^n$ we are left with $n^{2} 3^{1-n} x^{n}$ now ...
2
votes
2answers
33 views

“Convoluted” sum convergence: $\sum_{i=0}^{n} a_i \sum_{k=0}^{n-i} b_k$

Suppose $\sum_{j=0}^{\infty} a_j =a$, $\sum_{j=0}^{\infty} b_j =b$. Is it true (or under what conditions is it true) that: $$\lim_{n \rightarrow \infty}\sum_{i=0}^{n} \left( a_i \sum_{k=0}^{n-i} b_k ...
0
votes
0answers
36 views

Convergence of $\sin(x)/x$ using Dirichlets Test [duplicate]

How do you go about proving convergence of $$\sum_{k=1}^{\infty} \dfrac{\sin(k)}k$$ using Dirichlet's Test?
2
votes
1answer
39 views

Showing that a sum diverges

Suppose that $a_{j} \geq 0$ and that $\sum a_{j}$ diverges. Prove that $\sum\frac{a_{j}}{1+a{j}}$ diverges. The hint that is given is show that it if it converges $a_{j} \rightarrow 0$. I don't ...
2
votes
2answers
87 views

Interval of convergence of $\sum_{n=4}^\infty x^n/n^5$

Find the interval of convergence $$\sum_{n=4}^\infty x^n/n^5$$ I'm lost here. My intuition was to use the ratio test. $$\lim_{n \to \infty} \frac{x^{n+1}}{(n+1)^5} \times \frac{n^5}{x^n} $$ ...
2
votes
1answer
37 views

Convergent series and of positive integers and partial sums.

Let $\sum a_n$ be a convergent series of positive real numbers with sum $s$ and partial sums $s_n=a_1+a_2+\cdots+a_n$. Prove that $\sum na_n$ is convergent if and only if $\sum (s-s_n)$ is ...
2
votes
1answer
40 views

How do I calculate the sum of $\sum_{k=1}^{\infty}\frac{(2-x)^k}{2^k\cdot k}$ in every x in (0, 4)?

Well I've been trying to search for the appropriate derivative but I couldn't find it Thanks
0
votes
1answer
53 views

trigonometric summation

Taking into consideration the functions $$\sum_{t=0}^{n} \sin{(\theta + t \phi)}=\frac{\sin({\frac{(n+1)\phi}2})\sin{(\theta+\frac{n \phi}2)}}{\sin{(\frac{\phi}2)}}$$ and ...
3
votes
2answers
85 views

Convergence of integral of log and sum the mean of the logs

How can I show that the following limit converges and $L \in (0, +\infty)$? $\lim\limits_{n \to +\infty}\left( S_n - T_n\right)$, where $S_n = \int\limits_1^n \log x\, dx$, and $T_n = \sum_{k = ...
4
votes
4answers
157 views

What's wrong with $\sum_{i=0}^{\infty}x^i = \frac{1}{1-x}$

Set $$y=\sum_{i=0}^{\infty}x^i $$ Multiply both side by $x$, then we have $$yx=\sum_{i=0}^{\infty}x^{i+1}$$ Use the first one to minus the second one, we have $$y(1-x)=1$$ Then we have ...
2
votes
1answer
31 views

Where am I making my mistake? (intervals of convergence)

The sum is: \begin{align} \sum_{n=1}^{\infty} \frac{(x-6)^n}{(-8)^nn} \end{align} I end up with $|(x - 6)/8| < 1$ and therefore, $-8 < 6 - x < 8$ so $14 > x > -2$, but that gives me the ...
16
votes
0answers
654 views

Prove that sum is finite

Let $j \in \mathbb{N}$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Please help me to prove that the following sum is ...