Advanced High-School Mathematics

SECTION 5.2 Numerical Series 267

Fact 2: If Fact 1 is true, we still need to show that there is some

numberM such that

∑k

n=0

an≤M for allk.

Fact 3: Even when we have verified Facts 1 and 2, we still might

not (and usually won’t) know the actual limit of the infinite series

∑∞

n=0

an.

Warning about Fact 1: the requirement that limn→∞an = 0 is a
necessary but not sufficientcondition for convergence. Indeed, in

the above we saw that the series

∑∞

n=1

1

n

divergesbut that

∑∞

n=1

1

n^2

con-

verges.

Exercises

1. ApplyFact 1above to determine those series which definitely will
   not converge.

(a)

∑∞

n=0

n

n+ 1

(b)

∑∞

n=0

(−1)n

n

(c)

∑∞

n=2

lnn

n

(d)

∑∞

n=0

n

(lnn)^2

(e)

∑∞

n=1

sinn

n

(f)

∑∞

n=0

(−1)nsinn

(g)

∑∞

n=2

(−1)nn^2

2 n

(h)

∑∞

n=0

n!

2 n

(i)

∑∞

n=2

lnn

ln(n^2 + 1)

2. Occassionally an infinite series can be computed by using a partial
   fraction decomposition. For example, note that

1

n(n+ 1)

=

1

n

−

1

n+ 1

and so

∑∞

n=1

1

n(n+ 1)

=

∑

n=1

(

1

n

−

1

n+ 1

)

=

(

1 −

1

2

)

+

( 1

2

−

1

3

)

+

( 1

3

−

1

4

)

+···= 1.

Such a series is called a “telescoping series” because of all the

internal cancellations. Use the above idea to compute