Assignment:
Q1) a) Prove that
N
Σ 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) Σ n²/n³ +10³
n=1
∞
ii) Σ 1/tan-¹(n)
n=1
∞
iii) Σ(-1)n/In(n)
n=2
Q2) a) Express (1/n+1) - (1/n+3) as a single fraction
using this result, or otherwise, prove that
∞
Σ 1/(n² +4n+3) = ½{(3/2)-(1/n+2)-(1/n+3)}
n=0
Hence determine whether the infinite series
∞
Σ 1/(n² +4n+3)
n=0
is convergent or divergent, justify your determination
b) The series
∞
Σ ( n+7)/(4n-1)
n=1
is divergent. Explain in detail why this is the case.
Provide complete and step by step solution for the question and show calculations and use formulas.