MCS 351 ENGINEERING MATHEMATICS SOLUTION OF ...

1-3 Uniform Convergence :
1. Let
∞∑
(z − 2i) 2n and |z − 2i| ≤ 0.999. We know that
n=0
1
1 − z = 1 + z + z2 + z 3 + . . .
1
∞
1 − (z − 2i) 2 = ∑
(z − 2i) 2n = 1 + (z − 2i) 2 + (z − 2i) 4 + (z − 2i) 6 + . . .
n=0
Because |z − 2i| ≤ 0.999, this serie converges uniformly.
2. Let
∞∑
n=0
z 2n+1
(2n + 1)! and |z| ≤ 1010 .999. Since
sinh z =
∞∑
n=0
z 2n+1
(2n + 1)! = z + z3
3! + z5
5! + . . .
this Maclaurin series of sinh z converges uniformly on every bounded set.
3. Let
∞∑
n=0
π n
n 4 z2n and |z| ≤ 0.56. From ratio test, we have
lim L = lim
n→∞ n→∞
∣
π n+1
(n+1) 4
π n
n 4
∣ ∣ = lim
n→∞
and |z| ≤ 0.56, this serie converges uniformly on everywhere.
πn 4
(n + 1) 4 = π > 1,
9-11 Power series :
9. Let
∞∑ (z + 1 − 2i) n
4 n . By ratio test, we get
n=0
lim L = lim
n→∞ n→∞
∣
1
4 ( n+1)
1
4 n ∣ ∣ = lim
n→∞
then we can say that this serie converges uniformly on everywhere.
10. Let
∞∑
n=0
(z − i) 2n
. Since
(2n)!
cosh z =
∞∑
n=0
1
4 = 1 4 < 1,
z 2n
(2n)! = 1 + z2
2! + z4
4! + . . .
this Taylor series of cosh z with center i converges uniformly on every bounded set.
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