# Thread: Convergence of infinite series

1. ## Convergence of infinite series

Investigate the convergence of the following infinite series and find their
sums, where appropriate:

i)

ii)

iii)

Hint: Find explicit expressions for the partial sums SN.

2. Originally Posted by matty888
Investigate the convergence of the following infinite series and find their

sums, where appropriate:

i)

ii)

iii)

Hint: Find explicit expressions for the partial sums SN.

I will give you a hint...in part i there is a cancellation that can be made and in part ii and iii they are telescoping series

3. Originally Posted by matty888
Investigate the convergence of the following infinite series and find their

sums, where appropriate:

i)

ii)

iii)

Hint: Find explicit expressions for the partial sums SN.
Expanding a small bit on post #2:

i) $\frac{2^n + 3(5^n)}{5^{n+1}} = \left(\frac{1}{5}\right) \, \left( \frac{2^n}{5^n} \right) + \frac{3}{5}$. It's the second term that will cause the series to diverge (why?)

ii) Using partial fractions: $\frac{4}{(4n-3)(4n+1)} = \frac{1}{4x-3} - \frac{1}{4x+1}$.

For both ii) and iii), write out the first few terms of each series and note the cancelling of terms that you can do .....