Investigate the convergence of the following infinite series and find theirsums, where appropriate:
i)
ii)
iii)
Hint: Find explicit expressions for the partial sums SN.
Expanding a small bit on post #2:
i) $\displaystyle \frac{2^n + 3(5^n)}{5^{n+1}} = \left(\frac{1}{5}\right) \, \left( \frac{2^n}{5^n} \right) + \frac{3}{5}$. It's the second term that will cause the series to diverge (why?)
ii) Using partial fractions: $\displaystyle \frac{4}{(4n-3)(4n+1)} = \frac{1}{4x-3} - \frac{1}{4x+1}$.
For both ii) and iii), write out the first few terms of each series and note the cancelling of terms that you can do .....