Estimating the Value of a Series

Main idea

Most series can't be added up exactly, so you add the first $n$ terms and then put a fence around everything you left out, the remainder $R_n = s - s_n$ . Knowing how big the leftover can be is what turns an approximation into an answer you can trust.

Integral bound

\int_{n+1}^{\infty} {\htmlData{sym=1}{f(x)}}\,dx \le {\htmlData{sym=0}{R_n}} \le \int_n^{\infty} {\htmlData{sym=1}{f(x)}}\,dx

What the symbols mean

$R_n$
Remainder
The error s − sₙ: everything you didn't add.
$f(x)$
Positive decreasing function
The function behind the terms, with f(n) = aₙ.

Alternating series bound

|R_n| \le {\htmlData{sym=0}{b_{n+1}}}

What the symbols mean

$b_{n+1}$
First omitted term
The size of the next term you didn't add.

Ratio bound

R_n \le \frac{{\htmlData{sym=0}{a_{n+1}}}}{1 - {\htmlData{sym=1}{r}}}

What the symbols mean

$a_{n+1}$
First omitted term
The next term after the ones you added.
$r$
Ratio cap
A number below 1 that every later term-to-term ratio stays under.

Explanation

Estimating a series is simply adding up the first bunch of terms and then figuring out how far off you could be. The leftover part is called the remainder (it's what remains after you stop adding), and the whole game is putting an upper limit on it. That's useful because almost every series you meet in practice, like $\sum \frac{1}{n^3}$ , has no nice closed-form answer, yet you still need a number. The equation $s = s_n + R_n$ is just three pieces: $s$ (the true total), $s_n$ (what you actually added up) and $R_n$ (everything you skipped). It looks like bookkeeping, and that's really all it is.

Every test in this unit answered a yes-or-no question: does it converge? This lesson asks the follow-up an engineer actually cares about: to what, and how close am I? The nice surprise is that the tests you already know come with error bounds built in. The work is in bounding $R_n$ .

Integral bound (positive, decreasing terms). The same rectangles as in the integral test squeeze the tail between two integrals. Add them to $s_n$ and you get an interval that contains $s$ , often a very tight one.

Comparison. If $0 \le a_n \le b_n$ , the tail of $\sum a_n$ is at most the tail of $\sum b_n$ , which you may be able to compute (a geometric tail, for instance).

Alternating series. The error is at most the first omitted term, exactly as you saw in the alternating series lesson.

Ratio bound. If every ratio $\frac{a_{k+1}}{a_k}$ after $a_{n+1}$ stays at or below some $r < 1$ , the tail is at most the geometric series $a_{n+1}(1 + r + r^2 + \cdots) = \frac{a_{n+1}}{1-r}$ . It's the ratio test's argument again, this time keeping track of the numbers.

A common mistake. It's tempting to think that once a series converges, a modest number of terms will get you close. Different series converge at wildly different speeds. Getting three decimals of $\sum \frac{(-1)^{n+1}}{n}$ takes a thousand terms; $\sum \frac{(-1)^n}{n!}$ needs seven. Always check the bound instead of guessing.

See it

Estimating Σ 1/n³ by adding the first N terms and fencing in the rest with two integrals. With just one term the bounds run from 1.125 to 1.5, which is pretty loose. As you can see, they pinch together fast around the true value 1.2020569: the gap shrinks like 1/N³, so by N = 20 the two bounds are only about 0.0001 apart.

Worked examples

Example 1

Estimate $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^3}$ using $s_{10}$ , and bound the error.

1
Add the first ten terms: $s_{10} = 1 + \frac18 + \frac{1}{27} + \cdots + \frac{1}{1000} \approx 1.19753$ .

Example 2

How many terms of $\displaystyle\sum \frac{(-1)^{n+1}}{n}$ guarantee an error below $0.001$ ?

1
For an alternating series the error is at most the first term you leave out, so you need $b_{n+1} = \dfrac{1}{n+1} < 0.001$ , which means $n + 1 > 1000$ .

$n = 1000$ terms. This series converges slowly.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Bound the error when $s_{100}$ is used to approximate $\displaystyle\sum \frac{1}{n^2}$ , using $\displaystyle\int_n^{\infty} f(x)\,dx$ .

Problem 2+5 XP

How many terms of $\displaystyle\sum \frac{(-1)^{n+1}}{n^2}$ guarantee an error of at most $0.005$ ?

Problem 3+5 XP

What error bound do you get using the first 5 terms of $\displaystyle\sum \frac{(-1)^{n+1}}{n^2}$ ?

Problem 4+5 XP

How many terms of $\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}$ (starting from $n = 0$ ) guarantee an error below $0.001$ ?

Problem 5+5 XP

Using $s_{10} \approx 1.19753$ and the integral bounds from the example, estimate $\displaystyle\sum \frac{1}{n^3}$ to three decimals.

Problem 6+5 XP

The Stefan–Boltzmann law of thermal radiation involves $\sum_{n=1}^\infty\frac{1}{n^4}$ . If you add the first $N$ terms, the rest is at most $\int_N^\infty\frac{dx}{x^4}$ . What's the smallest $N$ that guarantees an error below $0.0005$ ?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Using only the first $5$ terms of $\sum\frac{1}{n^3}$ plus integral bounds on the rest, estimate the sum as the midpoint of the interval $s_5 + \int_6^\infty\frac{dx}{x^3} \le S \le s_5 + \int_5^\infty\frac{dx}{x^3}$ .

Where it's used

Scientific libraries compute functions like $e^x$ and the error function by summing series until a remainder bound falls below machine precision. The bounds in this lesson are what tell the code when to stop.

Why a Square Wave Has Finite PowerCalc II · Electrical Engineering

Strategy for Series Power Series

Estimating the Value of a Series

Main idea

Integral bound

\int_{n+1}^{\infty} {\htmlData{sym=1}{f(x)}}\,dx \le {\htmlData{sym=0}{R_n}} \le \int_n^{\infty} {\htmlData{sym=1}{f(x)}}\,dx

What the symbols mean

$R_n$
Remainder
The error s − sₙ: everything you didn't add.
$f(x)$
Positive decreasing function
The function behind the terms, with f(n) = aₙ.

Alternating series bound

|R_n| \le {\htmlData{sym=0}{b_{n+1}}}

What the symbols mean

$b_{n+1}$
First omitted term
The size of the next term you didn't add.

Ratio bound

R_n \le \frac{{\htmlData{sym=0}{a_{n+1}}}}{1 - {\htmlData{sym=1}{r}}}

What the symbols mean

$a_{n+1}$
First omitted term
The next term after the ones you added.
$r$
Ratio cap
A number below 1 that every later term-to-term ratio stays under.

Explanation

Comparison. If $0 \le a_n \le b_n$ , the tail of $\sum a_n$ is at most the tail of $\sum b_n$ , which you may be able to compute (a geometric tail, for instance).

Alternating series. The error is at most the first omitted term, exactly as you saw in the alternating series lesson.

See it

Worked examples

Example 1

Estimate $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^3}$ using $s_{10}$ , and bound the error.

1
Add the first ten terms: $s_{10} = 1 + \frac18 + \frac{1}{27} + \cdots + \frac{1}{1000} \approx 1.19753$ .

Example 2

How many terms of $\displaystyle\sum \frac{(-1)^{n+1}}{n}$ guarantee an error below $0.001$ ?

1
For an alternating series the error is at most the first term you leave out, so you need $b_{n+1} = \dfrac{1}{n+1} < 0.001$ , which means $n + 1 > 1000$ .

$n = 1000$ terms. This series converges slowly.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Bound the error when $s_{100}$ is used to approximate $\displaystyle\sum \frac{1}{n^2}$ , using $\displaystyle\int_n^{\infty} f(x)\,dx$ .

Problem 2+5 XP

How many terms of $\displaystyle\sum \frac{(-1)^{n+1}}{n^2}$ guarantee an error of at most $0.005$ ?

Problem 3+5 XP

What error bound do you get using the first 5 terms of $\displaystyle\sum \frac{(-1)^{n+1}}{n^2}$ ?

Problem 4+5 XP

How many terms of $\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}$ (starting from $n = 0$ ) guarantee an error below $0.001$ ?

Problem 5+5 XP

Using $s_{10} \approx 1.19753$ and the integral bounds from the example, estimate $\displaystyle\sum \frac{1}{n^3}$ to three decimals.

Problem 6+5 XP

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Where it's used

Why a Square Wave Has Finite PowerCalc II · Electrical Engineering