Prove that

(**Note:** I cannot get the bounds Apostol asks for. I prove a different set below. I cannot figure out if it is a mistake in the book or not.)

*Proof.* Using the algebraic identity, valid for ,

we obtain

Therefore, integrating term by term,

Furthermore, we have

Taking , we then have

From the inequality for this integral we then have

The integral evaluates to a number slightly higher than the upper bound in the book, so I guess he rounded to 6 decimals. If you take the first 4 terms up till x^12 and round the result, you get the upper bound in the book.