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.


Suppose first thatfor all




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


which is a standard convergent sequence thereforeconverges.