The series1+2^p/2!+3^p/3! + ....is
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Answer:
To show that the series 1+
2!
2
p
+
3!
3
p
+
4!
4
p
+...
is convergent for all values of p.
Here,
a
n
=
n!
n
p
We are using series ratio test
If there exists an N so that for all n≥N, a
n
=0
and L=
n→∞
lim
∣
∣
∣
∣
∣
a
n
a
n+1
∣
∣
∣
∣
∣
1 ) If L<1, then ∑a
n
converges
2) If L>1, then ∑a
n
diverges
3) If L=1, then the ratio test is inconclusive
n→∞
lim
∣
∣
∣
∣
∣
∣
∣
∣
∣
n!
n
p
(n+1)!
(n+1)
p
∣
∣
∣
∣
∣
∣
∣
∣
∣
n→∞
lim
∣
∣
∣
∣
∣
(n+1)n
p
(n+1)
p
∣
∣
∣
∣
∣
n→∞
lim
∣
∣
∣
∣
∣
n
p
(n+1)
p−1
∣
∣
∣
∣
∣
On applying limit, we get
L=lim
n→∞
∣
∣
∣
∣
∣
n
p
(n+1)
p−1
∣
∣
∣
∣
∣
L=0
Since, L<1
Hence, the given series converges
Step-by-step explanation:
Hope it's helpful to you
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