Sign up for your FREE 7-day trial.Get instant access to all the benefits of SparkNotes PLUS! Cancel within the first 7 days and you won't be charged. We'll even send you a reminder.

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Step 2 of 4

Choose Your Plan

Step 3 of 4

Add Your Payment Details

Step 4 of 4

Payment Summary

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thanks for creating a SparkNotes account! Continue to start your free trial.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

Suppose we have a function f (x), defined for all x≥1, which is positive and
decreasing. This function defines a sequence {f (n)} and a series

f (n) = f (1) + f (2) + ^{ ... }

Considering the following figure, we see that

f (2)≤f (x)dx

since a rectangle with height f (1) and width 1 is contained within the region below the
graph of f from 0 to 1.

Similarly,

f (3)≤f (x)dx

and so on. Thus we have

f (1) + f (2) + ^{ ... } + f (n)≤f (1) + f (x)dx

But the left side of this inequality is simply the nth partial sum for the series under
consideration. If

f (x)dx

is defined, then the partial sums are bounded, so the series converges.

This logic goes the other way too. As the figure below demonstrates,

f (1)≥f (x)dx

and so on.

Thus

f (1) + f (2) + ^{ ... } + f (n)≥f (x)dx

If does not exist, then the integral becomes
arbitrarily large for large n, as do the partial sums for the series. Therefore the series
f (n) does not converge. We summarize the results of this section in
the following statement.

If f is a positive, decreasing function defined for x≥1, then f (n) converges if and only if the limit

f (x)dx

exists. If so, denote this limit by L. Then the value to which the sum converges
satisfies the following inequality: