Articles of power series

Power series expansion

I recently had a problem. I know how to evaluate power series but I cannot seem to find an expansion for $\sqrt{x+1}$. I’ve tried differentiating it, in order to bring it in reciprocal form but that didn’t help. Due to the presence of square root, I cannot change it in the form of $1/(x+1)$. Kindly […]

Number of ways distribute 12 identical action figures to 5 children

Need a little help with this problem. Use generating functions to determine the number of different ways 12 identical action figures can be given to five children so that each child receives at most three action figures. So far I have that we are looking for the coefficient of $x^{12}$ and the generating function is […]

Translations AND dilations of infinite series

Sometimes, when working with infinite series, it’s useful to add “dilated” or “translated” versions of the infinite series, term by term, back to the original. There are ways of making this rigorous for each. If you want to add a “translated” copy of a series to the original one, you can: attach the series coefficients […]

Series expansion of $\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$?

How would I find the series expansion $\displaystyle\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$ so that it will turn into an infinite power series again??

Proof: $\lim \limits_{n\to\infty}(1+\frac{z}{n})^n = \exp(z)$

I define $\exp: \mathbb C \to \mathbb C$ as $z \mapsto \sum \limits_ {k=0}^{\infty}\frac{z^k}{k!}$. I would like to show that $\lim \limits_{n\to\infty}(1+\frac{z}{n})^n = \exp(z)$. I have a proof for the case $z \in \mathbb R$, but the proof assumes that $\lim \limits_ {n\to\infty}(1+\frac{z}{n})^n$ exists, which is easy to see if $z \in \mathbb R$, but […]

Solving a non-homogeneous differential equation via series solution

I have been set a question which asks me to solve: $$y”-2y=x^2e^{x^2}$$ by using a power series method. In trying to do it by brute force I end up with an non-homogeneous recurrence relation which is annoying to solve by hand. Is there a simple trick to solving this kind of non-homogeneous differential equation via […]

Estimating the series: $\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$

Any idea on how to estimate the following series: $$\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$$ where $a$ and $b$ are constant values. Greatly appreciate any respond.

Help on differential equation $y''-2\sin y'+3y=\cos x$

$y”-2\sin y’+3y=\cos x$ I’m trying to solve it by power series, but I just can’t find the way to get $\sin y’$. Is there any special way to find it?

Using Generating Functions to Solve Recursions

I have the recursion $A(n) = A(n-1) + n^2 – n$ with initial conditions $A(0) = 1$. I attempted to solve it using generating functions and I’m not quite sure I have it right, so I thought I might ask if anyone could take a look at my method so far. First I set up […]

Calculate sum of an infinite series

I have been struggling with this functional series. $$\sum_{n=1}^{\infty}{(-1)^{n-1}n^2x^n}$$ I need to calulate the sum.Any tips would be appreciated.