]> Exercise 1

# Definitions and Basics, Exercise 1

## Question

Use (2.1.1.10) and D'Alambert's test to prove that:

A power series can be integrated term by term within the interval of absolute convergence, without affecting the convergence in the following cases of power series:
1. $\sum _{n=0}^{\infty }{b}_{n}{x}^{n}$

2. $\sum _{n=0}^{\infty }{b}_{n}{x}^{2n+1}$

## Reminder

The D'alambert's test of the series with terms an states:

 $\text{If}\text{\hspace{0.17em}}\left\{\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}}\text{\hspace{0.17em}}|\frac{{a}_{n+1}}{{a}_{n}}|\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\rho \text{\hspace{0.17em}}<\text{\hspace{0.17em}}1\text{ }\text{the}\text{\hspace{0.17em}}\text{series converges absolutely}\\ \underset{n\to \infty }{\mathrm{lim}}\text{\hspace{0.17em}}|\frac{{a}_{n+1}}{{a}_{n}}|\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\rho \text{\hspace{0.17em}}>\text{\hspace{0.17em}}1\text{ }\text{the}\text{\hspace{0.17em}}\text{series diverges}\\ \underset{n\to \infty }{\mathrm{lim}}\text{\hspace{0.17em}}|\frac{{a}_{n+1}}{{a}_{n}}|\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\rho \text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{ }\text{the}\text{\hspace{0.17em}}\text{test}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{indecisive}\end{array}\right\}$ (1.3.5.22)

... rules for integration:

 $\int c\text{ }f\left(x\right)\text{d}x=c\int \text{ }f\left(x\right)\text{d}x\text{ }\left(\text{where}\text{ }c=\text{constant}\right)$ (2.1.1.8)
 $\int \left[{f}_{1}\left(x\right)+{f}_{2}\left(x\right)\right]\text{d}x=\int {f}_{1}\left(x\right)\text{d}x+\int {f}_{2}\left(x\right)\text{d}x$ (2.1.1.9)
 $\int {x}^{p}\text{d}x=\frac{{x}^{p+1}}{p+1}+C$ (2.1.1.10)

## Parts 1-4

Solution of question 1

1. For a power series written as $\sum _{n=0}^{\infty }{b}_{n}{x}^{n}$ the limit for the D'Alambert test is according to (1.3.5.22): $\underset{n\to \infty }{\mathrm{lim}}|\frac{{a}_{n+1}}{{a}_{n}}|=\underset{n\to \infty }{\mathrm{lim}}|\frac{{b}_{n+1}x}{{b}_{n}}|=\rho$
2. The integration of the series is obtained by use of the rules (2.1.1.8-10): $\int \text{d}x\sum _{n=0}^{\infty }{b}_{n}{x}^{n}=\sum _{n=0}^{\infty }{b}_{n}\int {x}^{n}\text{d}x=C+\sum _{n=0}^{\infty }\frac{{b}_{n}{x}^{n+1}}{n+1}$
3. The additive constant C from part 2 does not affect the convergence of the series.
4. The D'Alambert's test for this new series is: $\underset{n\to \infty }{\mathrm{lim}}|\frac{{b}_{n+1}x}{{b}_{n}}\frac{n}{n+1}|=\underset{n\to \infty }{\mathrm{lim}}\frac{n}{n+1}\text{\hspace{0.17em}}\underset{n\to \infty }{\mathrm{lim}}|\frac{{b}_{n+1}x}{{b}_{n}}|=\underset{n\to \infty }{\mathrm{lim}}|\frac{{b}_{n+1}x}{{b}_{n}}|$ exactly as the result of part 1, which proves the statement as required.

## Parts 5-7

Solution of question 2

1. For a power series written as $\sum _{n=0}^{\infty }{b}_{n}{x}^{2n+1}$ the limit of the D'Alambert's test is $\underset{n\to \infty }{\mathrm{lim}}|\frac{{a}_{n+1}}{{a}_{n}}|=\underset{n\to \infty }{\mathrm{lim}}|\frac{{b}_{n+1}{x}^{2}}{{b}_{n}}|$
2. The integration of the series yields $\int \text{d}x\sum _{n=0}^{\infty }{b}_{n}{x}^{2n+1}=\sum _{n=0}^{\infty }{b}_{n}\int {x}^{2n+1}\text{d}x=C+\sum _{n=0}^{\infty }\frac{{b}_{n}{x}^{2n+2}}{2n+2}$ where C as previously does not affect the convergence of the series.
3. The D'Alambert's test for this new series is: $\underset{n\to \infty }{\mathrm{lim}}|\frac{{b}_{n+1}{x}^{2}}{{b}_{n}}\frac{2n+2}{2n+4}|=\underset{n\to \infty }{\mathrm{lim}}|\frac{{b}_{n+1}{x}^{2}}{{b}_{n}}|\underset{n\to \infty }{\mathrm{lim}}|\frac{n+1}{n+2}|=\underset{n\to \infty }{\mathrm{lim}}|\frac{{b}_{n+1}{x}^{2}}{{b}_{n}}|$ in full agreement with part 5.

## Score

Questions 1 gives credit of 6 points.

Question 2 gives credit of 4 points.

By parts:

Parts 1,3,5,6 are worth 1 point each.
Parts 2,4,7 are worth 2 points each.