Q.o. - Calculus

Question

Read the section and make your own summary of the material 

Step-by-Step Solution

Answer

1) Comparison test : Let$\sum _{k=1}^{\infty }{a}_k and \underset{k=1}{\overset{\infty }{\sum b_k}}$  be two series with nonnegative terms such that $0\le a_k\le b_k$ for every positive integer k. If the series $\underset{k=1}{\overset{\infty }{\sum b_k}}$
converges , then the series$\sum _{k=1}^{\infty }{a}_k$  converges.

2) The limit comparison test

Let  $\sum _{k=1}^{\infty }{a}_k and \sum _{k=1}^{\infty }{b}_k$be two series with positive terms .

a) If$\underset{k\to \infty }{\lim } \frac{a_k}{b_k}=L$ , where L is any positive real number, then either both series converges or both series diverges.

b) If  $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=0$ and $\sum _{k=1}^{\infty }{b}_k$ converges, then $\sum _{k=1}^{\infty }{a}_k$ converges.

c) If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\infty$ and $\sum _{k=1}^{\infty }{b}_k$ diverges, then $\sum _{k=1}^{\infty }{a}_k$ diverges..

Given, The information in the section of book

1) Comparison test : Let $\sum _{k=1}^{\infty }{a}_k and \sum _{k=1}^{\infty }{b}_k$ be two series with nonnegative terms such that $0\le a_k\le b_k$ for every positive integer k. If the series $\sum _{k=1}^{\infty }{b}_k$ converges , then the series $\sum _{k=1}^{\infty }{a}_k$ converges.

step 2

2) The limit comparison test

Let $\sum _{k=1}^{\infty }{a}_k and \sum _{k=1}^{\infty }{b}_k$ be two series with positive terms .

a) If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=L$, where L is any positive real number, then either both series converges or both series diverges.

b) If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=0$ and $\sum _{k=1}^{\infty }{b}_k$ converges, then $\sum _{k=1}^{\infty }{a}_k$ converges.

c) If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\infty$ and $\sum _{k=1}^{\infty }{b}_k$ diverges, then $\sum _{k=1}^{\infty }{a}_k diverges$.