25 - Series Convergence Tests

Absolute Convergence

Let $(a_k)$ be a sequence then

$$\sum _{k=1}^{\infty }{a}_k\text{ converges absolutely}\text{ if }\sum _{k=1}^{\infty }|a_k|\text{ converges}$$

Ratio Test with Absolute Convergence

Let $(a_k)$ be a sequence of non-zero (real of complex) numbers
Assume that $\left|\frac{a_{k+1}}{a_k}\right|\to L\text{ as }k\to \infty$

1. If $0\le K<1$ then $\sum a_k$ converges absolutely and hence converges
2. If $K>1$ then $\sum a_k$ diverges

Similar to the ratio test if $L=1$ then you have to use another method

Proof (Not Detailed)

1. Apply Ratio Test to $(|a_k|)$
2. If $L>1$ then the Ratio Test applied to $(|a_k|)$ shows that

$$|a_j|\nrightarrow 0\text{ so }{a}_k\nrightarrow 0$$

Hence $\sum a_k$ diverges