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.

Short answer

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

Step-by-step solution

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 role="math" localid="1649136759276" $\sum _{k=1}^{\infty }{b}_k$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.