11.3) Determine whether the series is convergent or divergent. Please show a step by step solution.

Question

Determine whether the series is convergent or divergent. Please show a step by step solution.

Answer 1

1/1sqr1 + 1/2sqr2 + 1/3sqr3 + ...

= sum of 1/n^p where p=3/2, n = 1 through infinity

if p>1, the series converges, if p=1 or <1, the series diverges

in this example 3/2 >1 so the series converges

1 +1/2 +1/3 +1/4 + ... diverges because p=1, sum approaches infinity, this is the harmonic series

1 +1/2 +1/4 + 1/8 + ... converges because p=2>1, sum approaches 2

comparison test is compare a series to the harmonic series, if it is less than, it converges

1+ 1/2sqr2 + 1/3sqr3 ... is less than

1+ 1/2 + 1/3 ...

Answer 2

Use the Ratio Test.

L_x→∝ (1 / (n+1)^3/2) / (1 / (n)^3/2)

L_x→∝ n^3/2 / (n+1)^3/2

0

The series converges.