Mathematical Tools for Physics

2—Infinite Series 45

2.7 Useful Tricks
There are a variety of ways to manipulate series, and while some of them are simple they are probably not the
sort of thing you’d think of until you’ve seen them once. Example: What is the sum of

1 −

1

3

+

1

5

−

1

7

+

1

9

−···?

Introduce a parameter that you can manipulate, like the parameter you sometimes introduce to do integrals as in
Eq. (1.5). Consider the series with the parameterxin it.

f(x) =x−

x^3

3

+

x^5

5

−

x^7

7

+

x^9

9

−···

If I differentiate this with respect toxI get

f′(x) = 1−x^2 +x^4 −x^6 +x^8 −···

That looks a bit like the geometric series except that it has only even powers and the signs alternate. Is that
too great an obstacle? If 1 /(1−x)has only the plus signs, then 1 /(1 +x)alternates in sign. Instead ofxas a
variable, usex^2 , then you get exactly what you’re looking for.

f′(x) = 1−x^2 +x^4 −x^6 +x^8 −···=

1

1 +x^2

Now to get back to the original series, which isf(1)recall, all that I need to do is integrate this expression for
f′(x). The lower limit is zero, becausef(0) = 0.

f(1) =

∫ 1

0

dx

1

1 +x^2

= tan−^1 x

∣

∣

∣

∣

1

0

=

π

4

This series converges so slowly however that you would never dream of computingπthis way. If you take 100
terms, the next term is 1 / 201 and you can get a better approximation toπby using 22 / 7.
The geometric series is1 +x+x^2 +x^3 +···, but what if there’s an extra factor in front of each term?

f(x) = 2 + 3x+ 4x^2 + 5x^3 +···