-1
votes
3answers
77 views

Can there be more than one power series expansion for a function.

I guess the answer is NO, for polynomials. I know that there are more than one series expansion for every function. But I am talking about power series here. All Ideas are appreciated
3
votes
2answers
124 views

Calculate the value of $\sum\limits _{n=1}^{\infty }\:\dfrac{n}{2^n}$ [closed]

In a previous question it is asked to represent $f(x)=\dfrac{x}{1-x^2}$ as a power series. It gave me $\displaystyle\sum _{n=1}^{\infty \:}x\left(2x^2-x^4\right)^{n-1}$. Then they ask to use the last ...
3
votes
4answers
96 views

Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
-1
votes
2answers
34 views

Find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$

Let $\alpha$ and $\beta$ be two given constants, how to find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$, where $c_0 = {\alpha}^{\beta}$.
2
votes
2answers
77 views

how to multiply infinite power series

I am doing an assignment for my precalculus 2 class. I am expanding two infinite power series and multiplying them together to prove that $\exp(ax)\exp(by) = \exp(ax+by)$ I'm not sure what I am ...
1
vote
2answers
58 views

Simplifying ratio test with exponents $k+1$

Question: Find the interval and radius of convergence. $$\sum_{k=1}^\infty\frac{(x-1)^k(k^k)}{(k+1)^k} .$$ I applied the ratio test. ...
0
votes
3answers
51 views

How to get power by knowing the number and result

How to get power by knowing the number and result. For Example $$2^n = 8$$ how can i return the power $n$ by knowing number $2$ and result $8$ or $$4^n = 1024$$ how can i return the ...
1
vote
1answer
79 views

Practical significance of $e$ [duplicate]

We know, for example, the constant $\pi$ is the perimeter of a circle with diameter $1$ unit. In the similar manner how would we explain the constant $e$. I have searched a lot for it. But I couldn't ...
1
vote
1answer
65 views

Simplification of Binomial Expansion.

How $$(x+h)^n-x^n=nhx^{(n-1)}\text{ ?}$$ My attempt : $$ \begin{align} (x+h)^n-x^n & =nhx^{(n-1)} \\[8pt] & =\left[\sum_{k=0}^{n}\binom{n}{k}x^{(n-k)}h^k\right]-x^n \\[8pt] & = ...
4
votes
3answers
176 views

How to prove that $\frac{1}{(1-x)^3}$ is the generating function for the triangular numbers?

How to prove that $\dfrac{1}{(1-x)^3}$ is the generating function for the triangular numbers? The $n^{\text{th}}$ triangular number is defined as $T_n = \displaystyle{n+1 \choose 2}$. I used ...
6
votes
6answers
411 views

Why is $ \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$?

I am trying to see where this relationship comes from: $\displaystyle \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$ Does anyone have any special knowledge that me and my summer math teacher doesn't know ...
1
vote
3answers
49 views

Which one is the correct series expansion?

Is $$p^{n+1} = p^0+p^1+ \dots + p^n$$ or $$p^{n+1} = p^0\times p^1\times \dots \times p^n\text{ ?}$$ I am confused. please explain the correct one.
3
votes
3answers
244 views

the sum of a series

I am stuck on the computation of the following sum: $$\sum_{k=0}^{\infty} {\Big( {\frac{q}{k+1}} \Big)}^k ,$$ where $k$ is a natural number, and $0<q<1$.
12
votes
1answer
317 views

Solving a formal power series equation

I want to find a function $f(x,y)$ which can satisfy the following equation, $$\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = \exp \left[ \sum _{n=1} ^\infty ...
1
vote
1answer
62 views

Any dominance between these two functions?

Let $f\left(x\right):=e^{x}+e^{-x}+2$ and $g_{\beta}\left(x\right):=4e^{\beta x^{2}}$. Do there exist $a>0$ and $\beta>0$, such that $f\left(x\right)\le g\left(x\right)$ for all $x$, $0\le x\le ...
3
votes
4answers
537 views

Formula for calculating $\sum_{n=0}^{m}nr^n$

I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived. For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
0
votes
1answer
74 views

Manipulating Indices on Series

I have a series: $$ \sum_{n_1-l_1=0}^{\infty}\sum_{n_2-l_2=0}^{\infty}\sum_{n_3-l_3=0}^{\infty}a_{n_1-l_1,n_2-l_2,n_3-l_3}r^{n_1-l_1}s^{n_2-l_2}t^{n_3-l_3} $$ Which is equal to another series: $$ ...
0
votes
1answer
242 views

Sum of the polynomial roots raised to a power. How to prove?

Problem: If we have a polynomial $f$ with a derivative $f\,'$ and quotient $q$ function defined as: $$q(x)=\sum_{i=1}^{\infty}a_ix^{-i}=\frac{f\,'(x)}{f(x)},$$ and the roots of $f$ are ...
1
vote
1answer
219 views

How to find the sum? Based on logarithm function expansion

The problem: How to find the sum? $$-\sum_{i=1}^{\infty}\frac{(-x)^{i\; \bmod(k-1)}}{i}$$ Details: I tried find this sum using Mathematica ...