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# Infinite Series : Convergence and Divergence (8 Problems)

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1)
a) Prove that

N
&#8721; 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
&#8734;
i) &#8721; n²/n³ +10³
n=1

&#8734;
ii) &#8721; 1/tan&#8254; ¹(n)
n=1

&#8734;
iii) &#8721; (-1)&#8319;/In(n)
n=2

2)
a) Express (1/n+1) - (1/n+3) as a single fraction

using this result, or otherwise, prove that

&#8734;
&#8721; 1/(n² +4n+3) = ½{(3/2)-(1/n+2)-(1/n+3)}
n=0

Hence determine whether the infinite series

&#8734;
&#8721; 1/(n² +4n+3)
n=0
is convergent or divergent, justify your determination

b) The series
&#8734;
&#8721; ( n+7)/(4n-1)
n=1
is divergent. Explain in detail why this is the case