## Convergence For Sums of Sequences

The sum of a sequence is said to converge if

There are two very useful theorems for deciding whether or not the sum of a sequence converges.

The Ratio Test

If there existssuch thatforthenconverges. The test says nothing about sequences such thatwheremeans from below.

Proof:

Suppose first thatfor all

Hence

Example:

thereforeconverges.

The Comparison Test

We can prove convergence of divergence for some sequences by comparing the sequence with a 'standard' sequence, the sum of which which either converges or diverges.

For example supposeandthen

The proof is obvious.

Standard convergent sequences include

Standard divergent sequences include

Example:

which is a standard convergent sequence thereforeconverges.