# Infinite Series : Convergence and Divergence (8 Problems)

1)

a) Prove that

N

∑ 1/n(n+1) = 1- (1/N+1)

n=1

Hence, or otherwise, determine whether the following infinite series is convergent or divergent:

b) Determine whether each of these infinite series are convergent or divergent. Justify your answer

∞

i) ∑ nÂ²/nÂ³ +10Â³

n=1

∞

ii) ∑ 1/tan‾ Â¹(n)

n=1

∞

iii) ∑ (-1)ⁿ/In(n)

n=2

2)

a) Express (1/n+1) - (1/n+3) as a single fraction

using this result, or otherwise, prove that

∞

∑ 1/(nÂ² +4n+3) = Â½{(3/2)-(1/n+2)-(1/n+3)}

n=0

Hence determine whether the infinite series

∞

∑ 1/(nÂ² +4n+3)

n=0

is convergent or divergent, justify your determination

b) The series

∞

∑ ( n+7)/(4n-1)

n=1

is divergent. Explain in detail why this is the case

https://brainmass.com/math/real-analysis/infinite-series-convergence-and-divergence-8-problems-42068

#### Solution Summary

The convergence or divergence of series is investigated. The solution is detailed and well presented.