21 - Limit Form of Comparison Test

Limit Form of Comparison Test

Let $(a_k),(b_k)$ be real sequences of positive terms
Assume that $L>0$ such that

$$\frac{a_k}{b_k}\to L\text{ as }k\to \infty$$

Then

$$\sum a_k\text{ converges}⟺\sum b_k\text{ converges}$$

Proof

Since $\frac{a_k}{b_k}\to L$ as $k\to \infty$ and $\frac{L}{2}>0$
There is $K$ such that if $k\ge K$ then

$$\left|\frac{a_k}{b_{kk}}-L\right|<\frac{L}{2}\text{ hence }\frac{L}{2}<\frac{a_k}{b_k}<\frac{3L}{2}$$

1. $⟹$
   For $k\ge K$ then

$$0<a_k<\frac{3L}{2}{b}_k$$

So if $\sum b_k$ converges then $\sum \frac{3L}{2}{b}_k$ converges
Hence by Comparison Test then $\sum a_k$ converges
2) $⟸$
3For $k\ge K$ then $0<b_k<\frac{2}{L}{a}_k$ (with $L\ne 0$)
So if $\sum a_k$ converges then $\sum \frac{2}{L}{a}_k$ converrges
Hence by Comparison Test then $\sum b_k$ converges