Q. 7 - Calculus

Question

Provide a more general statement of the limit comparison test in which $\sum _{k=1}^{\infty }{a}_k$ and $\sum _{k=1}^{\infty }{b}_k$ are two series whose terms are eventually positive. Explain why your statement is valid. 

Step-by-Step Solution

Answer

The tail of the series determines whether the series will converge or diverge, the above statement is correct.

Step 1. Given information

Two series are $\sum _{k=1}^{\infty }{a}_k$ and $\sum _{k=1}^{\infty }{b}_k$ are eventually positive

Step 2. Verification

The limit comparison test for $\sum _{k=1}^{\infty }{a}_k$ and $\sum _{k=1}^{\infty }{b}_k$  are the series having positive terms the the following conditions may apply,

If $\underset{k\to \infty }{lim}\frac{a_k}{b_k}=L$,L must be positive number then it may be either converging or diverging. 

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

If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\infty$ then if $\sum _{k=1}^{\infty }{b}_k$ diverges then $\sum _{k=1}^{\infty }{a}_k$ diverges

The three sections of the above-mentioned limit comparison test remain unaltered.

Because the tail of the series determines whether the series will converge or diverge, the above statement is correct.