3.5 Power Series

Suppose $f(x)=\displaystyle\sum_{n=0}^\infty\left(\dfrac{3-x}{4}\right)^n$. What is $f(1)$?

Suppose $f(x)=\displaystyle\sum_{n=1}^\infty \dfrac{(x-5)^n}{n!+2}$. Give a power series representation of $f'(x)$.

Let $f(x)=\displaystyle \sum_{n=a}^\infty A_n(x-c)^n$ for some positive constants $a$ and $c$, and some sequence of constants $\{A_n\}$. For which values of $x$ does $f(x)$ definitely converge?

Let $f(x)$ be a power series centred at $c=5$. If $f(x)$ converges at $x=-1$, and diverges at $x=11$, what is the radius of convergence of $f(x)$?

(a) Find the radius of convergence of the series

(b) You are given the formula for the sum of a geometric series, namely:

Use this fact to evaluate the series in part (a).

Find the radius of convergence for the power series $\displaystyle\sum_{k=0}^\infty \frac{x^k}{10^{k+1}(k+1)!}$

Find the radius of convergence for the power series $\displaystyle\sum_{n=0}^\infty \frac{(x - 2)^n}{n^2+1}$ .

Consider the power series $\displaystyle\sum\limits_{n=1}^\infty \frac{(-1)^n(x+2)^n}{\sqrt{n}}$, where $x$ is a real number. Find the interval of convergence of this series.

Find the radius of convergence and interval of convergence of the series

Find the interval of convergence for the power series

Find all values $x$ for which the series $\displaystyle\sum_{n=1}^\infty\frac{(x+2)^n}{n^2}$ converges.

Find the interval of convergence for the following series.

Find, with explanation, the radius of convergence and the interval of convergence of the power series

Find the interval of convergence for the series $\displaystyle \sum_{n=1}^\infty (-1)^n n^2(x-a)^{2n}$ where $a$ is a constant.

Find the interval of convergence of the following series:

Find a power series representation for $\dfrac{x^3}{1-x}$.

Suppose $f'(x)=\displaystyle\sum_{n=0}^\infty \frac{(x-1)^{n}}{n+2}$, and $\displaystyle \int_5^x f(t)\dee{t}=3x+\displaystyle\sum_{n=1}^\infty \frac{(x-1)^{n+1}}{n(n+1)^2}$.

Give a power series representation of $f(x)$.

Determine the values of $x$ for which the series

converges absolutely, converges conditionally, or diverges.

(a) Find the power–series representation for $\displaystyle\int\frac{1}{1+x^3}\,\dee{x}$ centred at $0$ (i.e. in powers of $x$).

(b) The power series above is used to approximate $\displaystyle\int_0^{1/4}\frac{1}{1+x^3}\,\dee{x}$. How many terms are required to guarantee that the resulting approximation is within $10^{-5}$ of the exact value? Justify your answer.

(a) Show that $\displaystyle\sum_{n=0}^\infty nx^n =\frac{x}{(1-x)^2}$ for $-1<x<1$.

(b) Express $\displaystyle\sum_{n=0}^\infty n^2x^n$ as a ratio of polynomials. For which $x$ does this series converge?

Suppose that you have a sequence $\{b_n\}$ such that the series $\sum_{n=0}^{\infty}(1-b_n)$ converges. Using the tests we've learned in class, prove that the radius of convergence of the power series $\displaystyle\sum_{n=0}^{\infty}b_nx^n$ is equal to $1$.

Assume $\big\{a_n \big\}$ is a sequence such that $na_n$ decreases to $C$ as $n \rightarrow\infty$ for some real number $C > 0$

(a) Find the radius of convergence of $\displaystyle\sum\limits_{n=1}^\infty a_n x^n$ . Justify your answer carefully.

(b) Find the interval of convergence of the above power series, that is, find all $x$ for which the power series in (a) converges. Justify your answer carefully.

An infinitely long, straight rod of negligible mass has the following weights:

At what position is the centre of mass of the rod?

Let $f(x)=\displaystyle\sum_{n=0}^{\infty}A_n(x-c)^n$, for some constant $c$ and a sequence of constants $\{A_n\}$. Further, let $f(x)$ have a positive radius of convergence.

If $A_1=0$, show that $y=f(x)$ has a critical point at $x=c$. What is the relationship between the behaviour of the graph at that point and the value of $A_2$?

Evaluate $\displaystyle \sum_{n=3}^\infty \frac{n}{5^{n-1}}$.

Find a polynomial that approximates $f(x)=\log(1+ x)$ to within an error of $10^{-5}$ for all values of $x$ in $\left(0,\frac{1}{10}\right)$.

Then, use your polynomial to approximate $\log(1.05)$ as a rational number.

Find a polynomial that approximates $f(x)=\arctan x$ to within an error of $10^{-5}$ for all values of $x$ in $\left(-\frac{1}{4},\frac{1}{4}\right)$.