Why nonnegative terms?
Integral test Ordinary comparison Limit Comparison
Tom Lewis
Spring Semester
2015
Why nonnegative terms?
Integral test
Ordinary comparison Limit Comparison
Why nonnegative terms?
Integral test
Ordinary comparison
Limit Comparison

Why nonnegative terms?
Integral test Ordinary comparison Limit Comparison
The key idea
1.
The partial sums of a sequence of nonnegative numbers form a monotone increasing sequence.
2.
Therefore a series with nonnegative terms converges if and only if the associated sequence of partial sums is bounded above.
Why nonnegative terms?
Integral test Ordinary comparison Limit Comparison
Theorem (The integral test)
Let f : [ 1 ,
∞
) →
Then
R be a positive, monotone decreasing function.
n = 1 f ( n ) and
Z
∞ f ( x ) dx
1 converge or diverge simultaneously.

Why nonnegative terms?
Integral test Ordinary comparison Limit Comparison
Problem ( p -series test)
Show that
1 n = 1 n p diverges if p > 1 ; converges if p < 1 .
Why nonnegative terms?
Integral test Ordinary comparison Limit Comparison
Theorem (The ordinary comparison test)
Let { a n
} and { b n
} be nonnegative sequences, and suppose that there exists a positive integer M such that a n
6 b n for n > M
P
.
1.
If
P a n
2.
If b n diverges, then
P converges, then b
P n a n diverges.
converges.

Why nonnegative terms?
Integral test Ordinary comparison Limit Comparison
Problem
In each case determine whether the indicated series converges or diverges:
1.
n = 1 sin ( n )
.
n 2
2.
1 n = 1 n 1 + sin ( n )
.
Why nonnegative terms?
Integral test Ordinary comparison Limit Comparison
Theorem (The limit comparison test)
Let
{ a n
} be a sequence of nonnegative real numbers, let
{ b n
} be a sequence of positive real numbers, and assume that
a n
lim n → ∞ b n
= L .
There are three cases:
1.
If L is a finite positive real number, then the series b n converge or diverge simultaneously.
P b n
2.
If L = 0 , then the convergence of convergence of divergence of
P b n a n
, or the divergence of a n
P a n and implies the
3.
If L = +
∞
, then the convergence of convergence of divergence of
P a n b
.
.
n
P a n
, or the divergence of b n implies the

Why nonnegative terms?
Integral test Ordinary comparison Limit Comparison
Problem
In each case, determine whether the given series converges or diverges:
1.
2.
n = 1
2 n + 1
√ n 4 + 4 n = 1
3 n
+ 5 n
2 n + 6 n
.