Advanced High-School Mathematics

SECTION 5.2 Numerical Series 279

series (do this!). However, the terms decrease and tend to zero and so

by theAlternating Series Test

∑∞

n=0

(−1)nn

n^2 + 1

converges.

Exercises

1. Test each of the series below for convergence.

(a)

∑∞

n=1

(−1)n

n+ 2

n^2 + 10n

(b)

∑∞

n=2

(−1)n

lnn

(c)

∑∞

n=1

(−1)n

lnn

ln(n^3 + 1)

(d)

∑∞

n=1

( 1

n

−

1

n^2

)

(e)

∑∞

n=2

(−1)n

ln lnn

lnn

(f)

∑∞

n=1

((−1)n

1 +^1 n

)n

(g)

∑∞

n=1

(−1)n

√

n+

√

n+ 1

(h)

∑∞

n=1

(−2)n

n!

2. Determine whether each of the series above convergescondition-
   ally, convergesabsolutelyor diverges.


3. Prove that the series the improper integral

∫∞

−∞

sinx

x

dxconverges.^14

4. Prove that the improper integral

∫∞

0 cosx

(^2) dxconverges. (^15) (Hint:
try the substitutionu=x^2 and see if you can apply the Alternating
Series Test.)

5. Consider the infinite series

∑∞

n=0

n

2 n

, where eachnis±1. Show that

any real number x, − 2 ≤ x ≤ 2 can be represented by such a

series by considering the steps below:

(a) Write Σ =

∑∞

n=0

n

2 n

= Σ+−Σ−, where Σ+ is the sum of the

positive terms in Σ and where Σ−is−(negative terms in Σ).

(^14) In fact, it converges toπ.
(^15) This can be shown to converge to^1
2
...π
2.