Question

Discuss the convergence of series sigma 4,7,10.......(3n+1)/1,2,3.....n *x power of n

Answer 1

1
Lectures 11 - 13 : Infinite Series, Convergence tests, Leibniz’s theorem
Series : Let (an) be a sequence of real numbers. Then an expression of the form a1 +a2 +a3 +......
denoted by P∞
n=1 an, is called a series.
Examples : 1. 1 + 1
2 +
1
3 + .... or P∞
n=1
1
n
2. 1 + 1
4 +
1
9 + .... or P∞
n=1
1
n2
Partial sums : Sn = a1 + a2 + a3 + ...... + an is called the nth partial sum of the series P∞
n=1 an,
Convergence or Divergence of P∞
n=1 an
If Sn → S for some S then we say that the series P∞
n=1 an converges to S. If (Sn) does not
converge then we say that the series P∞
n=1 an diverges.
Examples :
1. P∞
n=1 log(
n+1
n
) diverges because Sn = log(n + 1).
2. P∞
n=1
1
n(n+1) converges because Sn = 1 −
1
n+1 → 1.
3. If 0 < x < 1, then the geometric series P∞
n=0 x
n
converges to 1
1−x
because Sn =
1−x
n+1
1−x
.
Necessary condition for convergence
Theorem 1 : If P∞
n=1 an converges then an → 0.
Proof : Sn+1 − Sn = an+1 → S − S = 0. ¤
The condition given in the above result is necessary but not sufficient i.e., it is possible that
an → 0 and P∞
n=1 an diverges.
Examples :
1. If | x | ≥ 1, then P∞
n=1 x
n diverges because an 9 0.
2. P∞
n=1 sinn diverges because an 9 0.
3. P∞
n=1 log(
n+1
n
) diverges, however, log(
n+1
n
) → 0.
Necessary and sufficient condition for convergence
Theorem 2: Suppose an ≥ 0 ∀ n. Then P∞
n=1 an converges if and only if (Sn) is bounded above.
Proof : Note that under the hypothesis, (Sn) is an increasing sequence. ¤
Example : The Harmonic series P∞
n=1
1
n
diverges because
S2
k ≥ 1 +
1
2
+ 2 ·
1
4
+ 4 ·
1
8
+ ... + 2k−1
·
1
2
k
= 1 +
k
2
for all k.
Theorem 3: If P∞
n=1 | an | converges then P∞
n=1 an converges.
Proof : Since P∞
n=1 | an | converges the sequence of partial sums of P∞
n=1 | an | satisfies the Cauchy
criterion. Therefore, the sequence of partial sums of P∞
n=1 an satisfies the Cauchy criterion. ¤
Remark : Note that P∞
n=1 an converges if and only if P∞
n=p
an converges for any p ≥ 1.