Those denominators are factorials:

4! = 4·3·2·1 = 24
3! = 3·2·1 = 6
2! = 2·1 = 2
1! = 1

also 0! is defined as 1.

Therefore the series:

 5 + 5 +  +  +  + ...

is really

  +  +  +  +  + ...

which is really:

  +  +  +  +  + ... +  + ...

So, the nth term is  , where we start n with 0.

We put a S in front of that with n=0 on the

bottom and  on the top, since the ellipsis "..." with
nothing after it indicates that it never ends.  So the sigma notation
is



By the way, you can use other letters for the "index" (or "dummy variable")
besides n.

These answers are just as good:

, , , , etc.

You can also start the index (or dummy variable) at 1 by subtracting 1
from the dummy variable after the sigma:

 is the same as 

In fact you can add any integer to the start of the dummy variable and subtract
the same integer from the dummy variable after the sigma.  For instance,  

 is the same as 
number 


Edwin