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.