Many infinite series cannot be added exactly, but calculus gives reliable ways to estimate their sums. The main idea is to add a finite number of terms, called a partial sum, and then bound the error from the terms left over. This matters because series are used to approximate functions, constants, probabilities, and physical quantities.
A good estimate always includes both a numerical value and a statement about how accurate it is.
For a convergent series, the remainder R_N = S - S_N measures the difference between the true sum S and the partial sum S_N. Alternating series often have a simple error bound based on the first omitted term, while positive decreasing series can often be bounded using improper integrals. These tools let you decide how many terms are needed before computing.
In applications, the goal is not just to calculate more terms, but to choose a method that proves the estimate is within a required tolerance.
Key Facts
- Partial sum: S_N = a_1 + a_2 + ... + a_N
- Remainder: R_N = S - S_N, so S = S_N + R_N
- Alternating series estimate: if a_n decreases to 0, then |R_N| <= a_{N+1}
- Integral test upper bound: for positive decreasing f with a_n = f(n), R_N <= integral from N to infinity of f(x) dx
- Integral test two-sided bound: S_N + integral from N+1 to infinity of f(x) dx <= S <= S_N + integral from N to infinity of f(x) dx
- To guarantee error less than epsilon, choose N so the error bound is less than epsilon
Vocabulary
- Infinite series
- An infinite series is a sum of infinitely many terms, usually written as sum from n = 1 to infinity of a_n.
- Partial sum
- A partial sum is the sum of the first N terms of a series and is used as an approximation to the full sum.
- Remainder
- The remainder is the difference between the exact infinite sum and a chosen partial sum.
- Alternating series
- An alternating series is a series whose terms switch sign, often written with a factor such as (-1)^n or (-1)^{n+1}.
- Error bound
- An error bound is a guaranteed maximum size for the possible error in an approximation.
Common Mistakes to Avoid
- Using S_N without an error bound, which is wrong because a decimal approximation alone does not prove how close it is to the true infinite sum.
- Applying the alternating-series error bound when the positive term sizes do not decrease to 0, which is wrong because the bound depends on both decreasing terms and a zero limit.
- Confusing the first omitted term with the last included term, which is wrong because the alternating-series error bound uses a_{N+1}, not a_N.
- Using the integral test bound with a function that is not positive and decreasing on the needed interval, which is wrong because the geometric area comparison may fail.
Practice Questions
- 1 Estimate the sum of the alternating series sum from n = 1 to infinity of (-1)^{n+1}/n using the first 5 terms. Give S_5 and an upper bound for the error.
- 2 For the series sum from n = 1 to infinity of 1/n^2, use the integral test remainder bound to find an N that guarantees the error after S_N is less than 0.01.
- 3 A student estimates a convergent positive series by computing S_10 but gives no error estimate. Explain why this is incomplete and describe what additional information would make the estimate reliable.