All Questions
7
questions
0
votes
1
answer
77
views
Please help me to find the sum of an infinite series. [duplicate]
Please help me to solve this problem. I need to find the sum of an infinite series:
$$
S = 1 + 1 + \frac34 + \frac12 + \frac5{16} + \cdots
$$
I tried to imagine this series as a derivative of a ...
- 21
0
votes
0
answers
33
views
Convergence of specific power series
I have to evaluate pointwise/uniform/total convergence of this series and I didn't quite understand how to do it.
$$\sum_{k=2}^{+\infty}{\ln k \over 2+\sin k}x^k$$
For pointwise convergence: it ...
- 415
0
votes
2
answers
169
views
Radius of convergence for $\sum_{n=0}^\infty n^nx^n$ and $\sum_{n=0}^\infty \frac {(-3)^n}{n}(x+1)^n$
How can one calculate the radius of convergence for the following power series:
$$\sum_{n=0}^\infty n^nx^n$$
and
$$\sum_{n=0}^\infty \frac {(-3)^n}{n}(x+1)^n$$
Regarding the first one I know ...
- 931
2
votes
1
answer
96
views
Solving ODE with power series
In some old notes, I found an exercise in which it was asked to solve this ODE, in a neighbourhood of $x_0=1$:
$$xy''(x) - 3y(x) = 2x^2$$
I tried to solve it but I'm getting stuck. Let me show you ...
- 341
0
votes
3
answers
735
views
Power of Series [duplicate]
In taking the power of a series
$$\left(\sum_{k=0}^{\infty} a_k x^k \right)^n =
\sum_{k=0}^{\infty} c_k x^k$$
do you know an expression for $c_k$ solely in terms of the coefficients $a_k$?
- 879
1
vote
2
answers
169
views
Sum of manipulated geometric series
Find the sum of $$\sum_{n=1}^{\infty} \frac{n^2}{2^n}$$
I know I need to manipulate the power series $\sum_{n=0}^{\infty}x^n$ with $x = \frac{1}{2}$, but I'm not sure how. Would differentiating it ...
- 121
0
votes
1
answer
2k
views
By applying term-wise differentiation and integration find the sum of the series $\sum_{k=1}^{\infty}\frac{x^k}{k}$
I need to find the sum of the following series: $$\sum_{k=1}^{\infty}\frac{x^k}{k}$$ on the interval $x\in[a,b], -1<a<0<b<1$ using term-wise differentiation and integration.
Can anyone ...
user184036