nth partial sums of series + telescoping series

Alright, so I'm having some conceptual trouble. I'll go ahead and post the homework problems but these are the concept in practice. I don't need all of the homework worked. Just pick and choose and help me see what's going on here. I only need to figure out what the nth partial sum is and how to reliably find it.

Homework problems, prefer help on one or two of the latter 3 but the first 2 are easier if you don't have the time.

"Find a formula for the nth partial sum of each series and use it to find the series' sum if the series converges."

2+ 2/3 + 2/9 + 2/27 + .... 2/3^(n-1) + ...

(Answer: s_n=((2(1-(1/3)^n)) / (1-(1/3) and it converges at 3.)

and

1/(2*3) + 1/(3*4) + 1/(4*5) + ... 1/((n+1)(n+2))

(Answer is: s_n=1/2 - 1/(n+2) converges at 1/2)

Then part 2 where I got my butt kicked for not understanding partial sums, in the homework,

"Find a formula for the nth partial sum of the series and use it to determine if the series converges or diverges. If it converges, find the sum."

**Note: I only need help finding s_n right now! **:)

Summation from n = 1 to infinity of (1/n - 1/(n+1)

( Answer: s_n= 1 - 1/n+1, converges at 1 )

and

Summation from n=1 to infinity of ln((n+1)^(1/2)) - ln(n^(1/2))

( Answer: s_n= ln((n+1)^(1/2)), diverges )

and

Summation from n=1 to infinity of 4/((4n-3)(4n+1))

( Answer: 1, no s_n given for answer )

Thanks guys! Pick and choose! Once I get it nailed down on one, the others will come easily!