## Estimating the Sum of a Series

Suppose is a convergent sequence of positive integers. Let

which is the error if you approximate using the first terms. From Theorem 6.3.2 we get the following.

Proposition 6.3.8 (Remainder Bound)   Suppose is a continuous, positive, decreasing function on and is convergent. Then

Proof. In Theorem 6.3.2 set . That gives

But

since is decreasing and .

Example 6.3.9   Estimate using the first terms of the series. We have

The proposition above with tells us that

In fact,

and we hvae

so the integral error bound was really good in this case.

Example 6.3.10   Determine if convergers or diverges. Answer: It converges, since

and converges.

William Stein 2006-03-15