Calculus BC: Series

If a _n is the n -th term of the series, then

For large n , this quotient is very close to 1/2 . Hence, for all but the first few integers n , the quotient will be ≤3/4 . Since we may disregard the first few terms when considering questions of convergence, the ratio test implies that the series converges.

Letting a _n equal the n th term of the series, we have

Since this ratio is clearly less than, say, 1/2 for all but the first few n , the ratio test implies that it converges.

If a _n is the n th term of the series, then

Since 1/r < 1 whenever r > 1 , the ratio test implies that the series converges.