Mathematical Tools for Physics

2—Infinite Series 37

For largen, the numerator is essentiallyn^3 and the denominator is essentially 5 n^5 , so for largenthis series is
approximately like
∑∞ 1
5 n^2
More precisely, the ratio of thenthterm of this approximate series to that of the first series goes to one as
n→∞. This comparison series converges, so the first one does too. If one of the two series diverges, then the
other does too.
Apply the ratio test to the series forex.

ex=

∑∞

0

xk/k! so

uk+1

uk

=

xk+1/(k+ 1)!

xk/k!

=

x

k+ 1

Ask→∞this quotient approaches zero no matter the value ofx. This means that the series converges for all
x.

Absolute Convergence
If a series has terms of varying signs, that should help the convergence. A series is absolutely convergent if it
converges when you replace each term by its absolute value. If it’s absolutely convergent then it will certainly be
convergent when you reinstate the signs. An example of a series that is convergent but not absolutely convergent
is ∞
∑

k=1

(−1)k+1

1

k

= 1−

1

2

+

1

3

−...= ln(1 + 1) = ln 2 (10)

Change all the minus signs to plus and the series is divergent. (Use the integral test.)
Can you rearrange the terms of an infinite series? Sometimes yes and sometimes no. If a series is convergent
but notabsolutelyconvergent, then each of the two series, the positive terms and the negative terms, is separately
divergent. In this case you can rearrange the terms of the series to converge to anything you want! Take the
series above that converges toln 2. I want to rearrange the terms so that it converges to

√

2. Easy. Just start
   adding the positive terms until you’ve passed

√

2. Stop and now start adding negative ones until you’re below
   that point. Stop and start adding positive terms again. Keep going and you can get to any number you want.

1 +

1

3

+

1

5

−

1

2

+

1

7

+

1

9

+

1

11

+

1

13

−

1

3

etc.