Calculus II · Unit 4A · lesson
Integral-Test Remainder Estimates
Bound series tails with improper integrals and convert a requested accuracy into a sufficient number of terms.
Section overview
Positive-series tests and error estimatesWhat this section is building
Bound series tails with improper integrals and convert a requested accuracy into a sufficient number of terms.
Direct comparison transfers inequalities; limit comparison transfers asymptotic scale; the integral test links sums to accumulated area.
Use a clean inequality when available, asymptotic comparison when ratios stabilize, and the integral test when a matching decreasing function is natural.
Reversing the direction needed to prove convergence or divergence, or forgetting an integral-test remainder condition.
Learning objectives
bound the tail of a convergent positive series and determine how many terms guarantee a target accuracy.
Integral-Test Remainder Estimates
Convergence without an error estimate is often not enough
Knowing that a series converges answers an existence question, but applications require a numerical guarantee. For a positive decreasing series, the tail after terms can be trapped between two improper integrals. These bounds convert a convergence proof into a practical stopping rule.
The remainder is positive for positive-term series. The lower and upper integral bounds differ only by shifting the starting point, and both become small as grows. To guarantee accuracy, solve the upper-bound inequality rather than calculating many partial sums and hoping their decimals have stabilized.
Convergence is only the first question
Once a positive decreasing series is known to converge, practical work asks how much remains after the first terms. The same rectangles used in the Integral Test trap the remainder between two improper integrals.
The estimate is useful because it turns an inaccessible infinite tail into an explicit inequality. To guarantee a requested accuracy, solve the upper-bound inequality for , then round upward. The bound is a certificate: it may not equal the true error, but it proves the error cannot exceed the stated amount.
The omitted rectangles are trapped by adjacent areas
For decreasing , each omitted rectangle lies below the area beginning one unit earlier and above the area beginning at its own right endpoint. Summing those comparisons yields
Remainder bounds
If the Integral Test hypotheses hold, then
How many terms for a cubic p-series?
For ,
To guarantee , require , so . Thus suffices.
Choose N from a target error
How many terms of guarantee error below ? For ,
Require
Since , taking guarantees the desired accuracy.
Use the correct endpoint for the upper bound
For a decreasing positive series, . Swapping the endpoints can turn a claimed guarantee into a guess.
u4a-integral_test_remainder_estimates-01Give an upper bound for the remainder of after terms.
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Integrate from to infinity.
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Find a remainder bound for .
How many terms ensure error below for ?
Explain why the bound may be larger than the actual error.
Use both lower and upper bounds to bracket the total sum from a partial sum.
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