Mathematical Tools for Physics

2—Infinite Series 35

As long asx < 1 this is precisely set up for the comparison test using

∑

nuKx

nas the series that dominates the
∑
nun. This test, theratio test is more commonly stated for positiveukas
If

uk+1

uk

≤x < 1 then the series

∑

uk converges (7)

This is one of the more commonly used convergence tests, not because it’s the best, but because it’s simple and
it works a lot of the time.

Integral Test
The integral test is another way to check for convergence or divergence. Iffis adecreasing positivefunction
and you want to determine the convergence of

∑

f(n), you can look at the integral

∫∞

dxf(x)and checkitfor
convergence. The series and the integral converge or diverge together.

1 2 3 4 5

f(1)

f(2)

f(3)

f(4)

f(x)

From the graph you see that the functionf lies between the tops of the upper and the lower rectangles.
The area under the curve offbetweennandn+ 1lies between the areas of the two rectangles. That’s the
reason for the assumption thatfis decreasing and positive.

f(n). 1 >

∫n+1

n

dxf(x)> f(n+ 1). 1

Add these inequalities fromn=kton=∞and you get

f(k) +f(k+ 1) +···>

∫k+1

k

+

∫k+2

k+1

+···=

∫∞

k

dxf(x)

> f(k+ 1) +f(k+ 2) +···>

∫∞

k+1

dxf(x)> f··· (8)