Download Worksheet 12 MATH 3283W Fall 2012 1. Show that the sequence a

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Transcript
Worksheet 12
MATH 3283W
Fall 2012
1. Show that the sequence an = sin(sin(. . . sin(1))) converges and find its limit.
{z
}
|
n
Hint. Recall that sin θ < θ for any angle θ > 0.
Definition. Given a sequence of real numbers (an ), we define a new sequence of partial
n
P
ak . If this new sequence (sn ) converges, we say that the series
sums (sn ) by the rule sn =
k=1
∞
P
ak converges. In this case the limit lim sn is called the sum of the series. Otherwise, the
k=1
series is said to diverge.
Example1 . Let an = rn for some fixed real number r and n ∈ N ∪ {0}. Then the series
∞
P k
r is called the geometric series and its n-th partial sum is equal to sn = 1 + r + · · · + rn =
k=0
1−rn+1
.
1−r
If |r| < 1, then this series converges and its sum is equal to
2. Find the formula for the partial sums of the series
converges.
3. Find the formula for the partial sums of the series
∞
P
k=2
∞
P
k=1
series converges.
1
Example 32.7
1
5k
1
1−r
+
1
7k
1
(k+1)(k+3)
and show that it
and show that the
∞
P
1
4. Find the formula for the partial sums of the series
and show that the series
k2 +3k+2
k=1
converges.
5. Let
∞
P
ak = a1 + a2 + · · · + ak + . . . be a convergent series with ak ≥ 0 for all k ∈ N.
k=1
Prove that the series
∞
P
a2k = a2 + a4 + · · · + a2k + . . . converges as well. Will it be
k=1
true without the assumption that ak ≥ 0?
2