Calculus II · Unit 4A · lesson

Integral-Test Remainder Estimates

Concept

Learning objectives

bound the tail of a convergent positive series and determine how many terms guarantee a target accuracy.

Integral-Test Remainder Estimates

Explanation

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 $N$ terms can be trapped between two improper integrals. These bounds convert a convergence proof into a practical stopping rule.

The remainder $R_N=S-s_N$ is positive for positive-term series. The lower and upper integral bounds differ only by shifting the starting point, and both become small as $N$ grows. To guarantee accuracy, solve the upper-bound inequality rather than calculating many partial sums and hoping their decimals have stabilized.

Bridge

Convergence is only the first question

Once a positive decreasing series is known to converge, practical work asks how much remains after the first $N$ terms. The same rectangles used in the Integral Test trap the remainder $R_N=S-s_N$ 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 $N$ , 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.

Proof idea

The omitted rectangles are trapped by adjacent areas

For decreasing $f$ , 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

\int_{N+1}^{\infty}f(x)\,dx\le R_N\le\int_N^{\infty}f(x)\,dx.

Concept

Remainder bounds

If the Integral Test hypotheses hold, then

\int_{N+1}^{\infty}f(x)\,dx\le R_N\le\int_N^{\infty}f(x)\,dx.

Guided walkthrough

How many terms for a cubic p-series?

For $\sum_{n=1}^{\infty}1/n^3$ ,

R_N\le\int_N^{\infty}x^{-3}\,dx=\frac1{2N^2}.

To guarantee $R_N<0.001$ , require $1/(2N^2)<0.001$ , so $N>\sqrt{500}$ . Thus $N=23$ suffices.

Worked example

Choose N from a target error

How many terms of $\sum_{n=1}^{\infty}1/n^4$ guarantee error below $10^{-4}$ ? For $f(x)=x^{-4}$ ,

R_N\le\int_N^\infty x^{-4}\,dx=\frac{1}{3N^3}.

Require

\frac{1}{3N^3}<10^{-4}, \qquad N^3>\frac{10^4}{3}.

Since $15^3=3375>3333.\overline3$ , taking $N=15$ guarantees the desired accuracy.

Common mistake

Use the correct endpoint for the upper bound

For a decreasing positive series, $\int_{N+1}^{\infty}f\le R_N\le\int_N^{\infty}f$ . Swapping the endpoints can turn a claimed guarantee into a guess.

Interactive checku4a-integral_test_remainder_estimates-01

Give an upper bound for the remainder of $\sum1/n^3$ after $N$ terms.

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Show hint

Integrate $x^{-3}$ from $N$ to infinity.

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Exercise

Find a remainder bound for $\sum1/n^2$ .

Exercise

How many terms ensure error below $10^{-4}$ for $\sum1/n^2$ ?

Exercise

Explain why the bound may be larger than the actual error.

Exercise

Use both lower and upper bounds to bracket the total sum from a partial sum.

After the explanation

Use the section idea

Reading lens

Compare positive terms by long-run size and select a benchmark whose convergence behavior is already known.

Mental model

Direct comparison transfers inequalities; limit comparison transfers asymptotic scale; the integral test links sums to accumulated area.

Decision

Use a clean inequality when available, asymptotic comparison when ratios stabilize, and the integral test when a matching decreasing function is natural.

Common trap

Reversing the direction needed to prove convergence or divergence, or forgetting an integral-test remainder condition.

Check yourself

Does your benchmark support the conclusion in the direction you claim, and are all hypotheses stated?

Source & rights

Original instruction with traceable references.

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