Calculus: Analytic Geometry and Calculus, with Vectors

12.1 Definitions and basic theorems 595

and hence must be convergent. Try to find simple reasons why $ <s <.

If more time is available, show that s = T. Remark: Proof that s = can be

based upon the identity

1 _ (^1) 1(p+1)-(p-1)
(p - 1)P(p + 1) p(p2 - 1) - 2 p(p2 - 1)

2L(p11)p p(p+1)

(^15) With the possibility of using consequences of the facts that

+ ++ + ... _ ,

(^1111)

1.22.3+34+4F5+ =1,

l+x+x2+x3+ ... =^1

1 - x (lxI < U.

when these things are helpful, tell whether and why the following series are con-
vergent or divergent:

(a)+12 4+8+16+...

(b) 1+22+42+32+162+ ...

(^02) 12 22 32 42
(e) 1222 + 22:32 + 32-42 + 4252 + 52:62 +
(d) 1 + +0+ 1 +. .+
(e) i+3+W+ +$+
(f)
(3) 102 + 213 + 324 + 435 + 5 +

(h)1+33+33+:3 +33+

(^1) 1 1 1 1
1+12+1+22+1+32+1+42+1+52+
log 2 log 3 log 4 log 5 log 6
(.1) 2 + 3 + 4 + 5 + 6 +
sin x sin 2x sin 3x sin 4x sin 5x
(k) 12 + 22 + 32 + 42 + 52 +
Hint: The comparison test is important. Seek a convergent series that dominates
your series (so you will know your series is convergent) or seek a divergent series
which your series dominates (so you will know that your series cannot be con-
vergent and must be divergent).
16 Prove that
to
2 1 1Ukflkl < 11412 + I Iakl2
k-1 ks1 k-1