All Questions
41
questions
1
vote
1
answer
89
views
How do I evaluate $\sum_{k=1}^nk^pr^k=?$
For this entire post, we have $r\ne1$, $n\in\mathbb N$. For the first half, $p\in\mathbb N$, and at the end $p\in\mathbb Q$.
It is well known that
$$\sum_{k=1}^nr^k=\frac{1-r^{n+1}}{1-r}$$
And
$$\...
3
votes
2
answers
74
views
Finding Exact Values of Specific Infinite Series
Prove that $\Sigma_{n=1}^{\infty}(n/2^n)=2$ and that $\Sigma_{n=1}^{\infty}(n^2/2^n)=6$.
Thoughts:
I have a feeling that if someone shows me how to do one, I'll be able to figure out the other. So ...
user322548
3
votes
6
answers
269
views
Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent?
Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent?
I tried ratio test but didn't seem to work, and I also know that the limit goes to zero, but I can't say its convergence because then....
- 33
0
votes
2
answers
1k
views
showing that the partial sums of $ \log(j) = n\log(n) - n + \text{O}(\log(n))$
I'm trying to show that the partial sums of $\log(j) = n\log(n) - n + \text{O}(\log(n))$
I know that $$\int_1^n\log(x)dx = n\log(n) - n + 1$$
so that this number is pretty close to what I want.
Now ...
- 13
4
votes
5
answers
338
views
Power series summation [closed]
Trying to find the sum of the following infinite series:
$$ \displaystyle\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{(2n-1)3^{n-1}}$$
Any ideas on how to find this sum?
- 277
0
votes
1
answer
70
views
How to calculate this sum?
Let $x_1,\cdots,x_k$ be numbers between 0 and 1. Then is it possible to get explicit expression for the following sum:$$\sum_{n_1,\cdots,n_k\geq 1} x_1^{n_1}\times C_{n_1+n_2}^{n_2}\times x_2^{n_2}\...
- 693
1
vote
1
answer
78
views
Trying to understand a power series example from Advanced Calculus by Taylor
Example 2 from 21.1 in the book, Find an expansion in powers of $x$ of the function
$$
f(x) = \int_{0}^{1} \frac{1-e^{-tx}}{t}dt
$$
and use it to calculate $f(1/2)$ approximately.
I ...
- 11
0
votes
2
answers
462
views
Proof for multiplication of two power series
Prove that $(\sum_{k=0}^\infty u^k)^2=\sum_{k=0}^\infty (k+1)u^k$ when |u|<1.
This is a proof I need for a larger proof I was doing. I am stuck on this, so I was not able to make any notable ...
- 211
4
votes
1
answer
4k
views
Exponential series is cosh(x), how to show using summation?
I want to show that $$\cosh(x) = \sum_{n=0}^{\infty} \frac{(x)^{2n}}{(2n)!}
$$
I know that $cosh(x) = \frac{exp(x)+exp(-x)}{2}$ but i cant seem to get there from the original series. I know that $$...
- 635
0
votes
1
answer
2k
views
By applying term-wise differentiation and integration find the sum of the series $\sum_{k=1}^{\infty}\frac{x^k}{k}$
I need to find the sum of the following series: $$\sum_{k=1}^{\infty}\frac{x^k}{k}$$ on the interval $x\in[a,b], -1<a<0<b<1$ using term-wise differentiation and integration.
Can anyone ...
user184036
0
votes
3
answers
165
views
Why $\sum_{n=0}^{\infty}(n+1)5^nx^n=\frac{1}{(1-5x)^2}$
Why $$\sum_{n=0}^{\infty}(n+1)5^nx^n=\frac{1}{(1-5x)^2}?$$
I know that $\sum_{n=0}^{\infty}x^n=\dfrac{1}{1-x}$, so by the same token, $\sum_{n=0}^{\infty}5^nx^n=\dfrac{1}{1-5x}$.
Thus
$$
\left(\...
- 12.8k