Mathematical Tools for Physics

2—Infinite Series 31

Some other common series that you need to know are power series for elementary functions:

ex= 1 +x+

x^2

2!

+··· =

∑∞

0

xk

k!

sinx=x−

x^3

3!

+··· =

∑∞

0

(−1)k

x^2 k+1

(2k+ 1)!

cosx= 1−

x^2

2!

+··· =

∑∞

0

(−1)k

x^2 k

(2k)!

ln(1 +x) =x−

x^2

2

+

x^3

3

−··· =

∑∞

1

(−1)k+1

xk

k

(3)

(1 +x)α= 1 +αx+

α(α−1)x^2

2!

+···=

∑∞

k=0

α(α−1)···(α−k+ 1)

k!

xk

Of course, even better than memorizing them is to understand their derivations so well that you can derive
them as fast as you can write them down. For example, the cosine is the derivative of the sine, so if you know
the latter series all you have to do is to differentiate it term by term to get the cosine series. The logarithm of
(1 +x)is an integral of 1 /(1 +x)so you can get it’s series from that of the geometric series. The geometric
series is a special case of the binomial series forα=− 1 , but it’s easier to remember the simple case separately.
You can express all of them as special cases of the general Taylor series.

What is the sine of 0. 1 radians? Just use the series for the sine and you have the answer, 0.1, or to more
accuracy, 0. 1 − 0. 001 /6 = 0. 099833

What is the square root of 1.1?

√

1 .1 = (1 +.1)^1 /^2 = 1 +^12. 0 .1 = 1. 05

What is 1/1.9? 1 /(2−.1) = 1/[2(1−.05)] =^12 (1 +.05) =.5 +.025 =. 525 from the first terms of the
geometric series.

What is^3

√

1024?^3

√

1024 =^3

√

1000 + 24 =^3

√

1000(1 + 24/1000) =

10(1 + 24/1000)^1 /^3 = 10(1 + 8/1000) = 10. 08