Calculus How To

Remainder of a Series: Step by Step Example, How to Find

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The remainder of a series, Rn = s – sn, is the difference between the nth partial sum(sn and the infinite (complete) sum (s) of the series.

Remainders are used to compare series in tests for convergence. For example, you could use the Integral Test to verify that a series converges. The remainder is the error in approximating the infinite sum with partial sums (Maslanka, 2020).

The remainder of the series is sometimes called the “error of the series.”

Remainder of a Series Example

In general, you estimate the remainder with a series of the first n sums. For example, you could use the first 10 terms, the first 12, or the first 29.

Example question: Use the first 10 terms to find the remainder of a series defined by:
remainder of a series 1

Step 1: Find the value for the first term. The terms start at n = 1 (stated at the bottom of the sigma notation). So, plugging in “1” to the formula, we get:
remainder of a series 2

Step 2: Find the value for the remaining terms. Keep going until you reach the stated number (10 terms).
remainder of a series 3
Note :For space here, I’ve just written 1, 2, 3, and 10 but (depending on your professor or exam), you may have to write out all of the values in between.

Step 3: Add up all of your terms from Step 2. To 3 decimal places, this example sums to ≈ 1.249.

That’s it!

Related Articles

Click the bold item to go to the article:

  1. Integral Test Remainder: Gives an upper bound and lower bound that demonstrates how close a remainder is to the actual series sum.
  2. Alternating Series Remainder: An alternating series remainder is the difference between our estimation of the series and the actual value.
  3. Taylor Series Remainder: Answers the question “how many degrees is good enough?”.


Maslanka, D. The Integral Test and Remainder Theorem. Math 152 Lecture Notes #7. Retrieved August 1, 2020 from:
Ikenaga, B. (2020). The Remainder Term. Retrieved August 1, 2020 from:
Kaplan Test Prep. (2020). AP Calculus BC Prep Plus 2020 & 2021. Simon & Schuster Digital Sales Inc.

Stephanie Glen. "Remainder of a Series: Step by Step Example, How to Find" From Calculus for the rest of us!

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