Find a Taylor series using the definition

e.g. “ Using the definition of a Taylor series, find the Taylor series for \dots centered at \dots ”

1
$Differentiate a few times and spot the pattern$ $f^{(n)}(x)$
2
$f^{(n)}(a)$
3
$\sum \dfrac{f^{(n)}(a)}{n!}(x - a)^{n}$ $If asked for the radius, run the ratio test on the result.$

Remember: Derivatives at the center, over n factorial.

See it in a problem3 worked examples

f(x) = 2^{x}

1
$Differentiate a few times and spot the pattern$ $f'(x) = 2^{x}\log 2$
2
$f^{(n)}(a)$ $f^{(n)}(0) = (\log 2)^{n}$
3
$\sum \dfrac{f^{(n)}(a)}{n!}(x - a)^{n}$ $\sum_{n=0}^{\infty} \dfrac{(\log 2)^{n}}{n!}x^{n}$

2^{x} = \displaystyle\sum_{n=0}^{\infty} \dfrac{(\log 2)^{n}}{n!}x^{n}

Common mistakes3

Writing $x^{n}$ instead of $(x - a)^{n}$ when the center is not $0$ .
Forgetting the $n!$ in the bottom.
Losing the sign pattern: at $n = 0$ , $(-1)^{n}$ is $+$ and $(-1)^{n+1}$ is $-$ .

2

Find a Maclaurin series by substituting into a known series

x^{3}e^{-x^{2}}

1
$Pick the known series that matches the outside function$
2
$x$ $x$
3
$Multiply by any outside factor and combine powers$
4
$State where it converges$ $u$

Remember: Known series, swap in, multiply out.

See it in a problem3 worked examples

x^{3}e^{-x^{2}}

1
$Pick the known series that matches the outside function$ $e^{u} = \sum_{n=0}^{\infty} \dfrac{u^{n}}{n!}$
2
$x$ $u = -x^{2}$
3
$Multiply by any outside factor and combine powers$ $x^{3}e^{-x^{2}} = \sum_{n=0}^{\infty} \dfrac{(-1)^{n}x^{2n+3}}{n!}$
4
$State where it converges$ $x$

\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^{n}x^{2n+3}}{n!}

Common mistakes3

Substituting into $x^{n}$ but not the whole term: $(3x)^{n} = 3^{n}x^{n}$ , not $3x^{n}$ .
Combining powers wrong: $x^{3} \cdot x^{2n} = x^{2n+3}$ , not $x^{6n}$ .
Differentiating from scratch when a substitution does it in one line.

3

Find a high-order derivative at a point using a Taylor series

f^{(20)}(3)

1
$f$
2
$(x - a)^{n}$ $0$
3
$n!$

Remember: Coefficient times n factorial.

See it in a problem3 worked examples

f(x) = x\,e^{x^{2}}

1
$f$ $f(x) = x\sum_{k=0}^{\infty} \dfrac{x^{2k}}{k!} = \sum_{k=0}^{\infty} \dfrac{x^{2k+1}}{k!}$
2
$(x - a)^{n}$ $x^{7}$
3
$n!$ $f^{(7)}(0) = 7! \cdot \dfrac{1}{6} = 840$

f^{(7)}(0) = 840

Common mistakes2

Forgetting the $n!$ : the coefficient is $\dfrac{f^{(n)}(a)}{n!}$ , not $f^{(n)}(a)$ .
Matching the summation index instead of the power: in $\sum x^{2k+1}$ , the $x^{7}$ term has $k = 3$ , not $k = 7$ .

4

Find the sum of a series by recognizing a Taylor series

e^{x}

1
$Match the terms to a known series$
2
$x$
3
$Adjust for a different starting index$ $e^{x}$
4
$x$

Remember: Find the twin series, then plug in.

See it in a problem3 worked examples

\displaystyle\sum_{n=1}^{\infty} \dfrac{2^{n}}{n!}

1
$Match the terms to a known series$ $\dfrac{x^{n}}{n!}$
2
$x$ $x = 2$
3
$Adjust for a different starting index$ $e^{x}$
4
$x$ $e^{2} - 1$

e^{2} - 1

Common mistakes3

Missing the start: $\sum_{n=1}^{\infty} \dfrac{x^{n}}{n!} = e^{x} - 1$ , not $e^{x}$ .
Plugging in an $x$ outside the interval: $\log(1 + x)$ only works for $-1 < x \leq 1$ .
Writing $\left(\tfrac{\pi}{3}\right)^{2n}$ as $\dfrac{\pi^{2n}}{3^{n}}$ . It is $\dfrac{\pi^{2n}}{9^{n}}$ .

5

Evaluate a limit using Taylor series

\displaystyle\lim_{x \to 0} \dots

1
$Replace each function by the first few terms of its series$
2
$Cancel, and keep the lowest surviving power on top and bottom$ $Add terms until each side has a nonzero term.$
3
$x \to 0$

Remember: Expand, cancel, lowest power wins.

See it in a problem3 worked examples

\displaystyle\lim_{x \to 0} \dfrac{x - \arctan x}{x^{3}}

1
$Replace each function by the first few terms of its series$ $\arctan x = x - \dfrac{x^{3}}{3} + \dfrac{x^{5}}{5} - \cdots$
2
$Cancel, and keep the lowest surviving power on top and bottom$ $x - \arctan x = \dfrac{x^{3}}{3} - \dfrac{x^{5}}{5} + \cdots$
3
$x \to 0$ $\dfrac{x - \arctan x}{x^{3}} = \dfrac{1}{3} - \dfrac{x^{2}}{5} + \cdots \to \dfrac{1}{3}$

\dfrac{1}{3}

Common mistakes2

Stopping the series too early: if everything cancels, you needed more terms.
Dropping a cross term when multiplying two series. Keep every power up to the one you need.

6

Approximate a number or integral with a Taylor series

10^{-4}

1
$Write the series, integrating term by term if needed$
2
$Plug in the value to get a series of numbers$
3
$Bound the error for each number of terms you try$ $0$
4
$Stop once the bound beats the tolerance; add up the terms$

Remember: Add terms until the error bound is small enough.

See it in a problem3 worked examples

\displaystyle\int_0^{1/2} \sin(x^{2})\,dx

1
$Write the series, integrating term by term if needed$ $\sin(x^{2}) = \sum_{n=0}^{\infty} \dfrac{(-1)^{n}x^{4n+2}}{(2n+1)!}$
2
$Plug in the value to get a series of numbers$ $\dfrac{1}{24}$
3
$Bound the error for each number of terms you try$ $\dfrac{1}{5376} \approx 1.9 \times 10^{-4}$
4
$Stop once the bound beats the tolerance; add up the terms$ $\dfrac{1}{24} - \dfrac{1}{5376} = \dfrac{223}{5376}$

\dfrac{223}{5376} \approx 0.04148

Common mistakes3

Using the first term you leave out as the bound when the series is not alternating. Then you need the remainder.
Bounding $e^{c}$ with the very number you are computing. Use a known bound such as $e < 3$ .
Miscounting: the terms $n = 0$ to $n = 2$ are three terms.

Taylor & Maclaurin Series

Find a Taylor series using the definition

Find a Maclaurin series by substituting into a known series

Find a high-order derivative at a point using a Taylor series

Find the sum of a series by recognizing a Taylor series

Evaluate a limit using Taylor series

Approximate a number or integral with a Taylor series

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Navigation

Find a Taylor series using the definition

Find a Maclaurin series by substituting into a known series

Find a high-order derivative at a point using a Taylor series

Find the sum of a series by recognizing a Taylor series

Evaluate a limit using Taylor series

Approximate a number or integral with a Taylor series