Question

Which of the following statements about the series
(-1)"
it un
n
is true?
A
The series converges absolutely.
B
The series converges conditionally.
C
The series converges but neither conditionally nor
absolutely
D
The series diverges.​

Answer 1

answer

option A is correct

as we can see ( -1)*1/n is convergence

Answer 2

Concept-

Convergent sequence is when through some terms you achieved a final and constant term as n approaches infinity and divergent sequence is that in which the terms never become constant , they continue to increase or decrease and they approach to infinity or - infinity as n approaches to infinity.

Given-

The series is given as ∑n=1  (-1)ⁿ / 1 + √n

Find-

The above given series will converge or diverge?

Solution-

If ∑aₙ converges but ∑|aₙ| diverges, then ∑aₙ converges conditionally.

Let ∑n=1  (-1)ⁿ/ 1+√n = ∑ aₙ ;        aₙ = (-1)ⁿ / 1 + √n

bₙ = 1 / 1 + √n , which alternating series , with

bₙ = 1 / 1 + √n > 0 and monotonic decreasing sequence.

And, lt. n→∞ bₙ = lt. n→∞ 1/1+√n = 1/∞ = 0

∴ By Alternating series -test  ,  ∑n=1  aₙ = ∑n=1  (-1)ⁿ. bₙ =

∑n=1  (-1)ⁿ. 1/1+√n  converges

Now, |aₙ| = | (-1)ⁿ / 1+√n|  =  1/1+√n = bₙ

Take Cₙ = 1/√n = 1/n^1/2   so that  ∑n=1 Cₙ = ∑n=1  1/√n  diverges by P- test

∴ lt. n→∞  bₙ / Cₙ = lt. 1/1+√n ×√n/1 = lt. n→∞ 1/1/√n+1 = 1/0+1 = 1 ≠ 0,∞

So, ∑n=1  bₙ and ∑n=1  Cₙ behave alike.

So, ∑n=1  bₙ = ∑n=1  |aₙ|

⇒ ∑n=1  1/1+√n  diverges

Therefore, the series converges conditionally.

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