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Help of summation of series

Hi,

I've been studying 'The Foundations Of Mathematics' by Ian Steward and David Tall and although I'm making progress some examples I still can't do some questions even going back a few chapters. Anyways the thing that is bugging me is that I still don't know how to work with limits particularly well even though it was covered early in the book. I can work out a limit of a sequence (finite and infinite) thanks to some youtube help. However I can't calculate the sum of a series, I've tried youtube and wikipedia but I don't quite understand whats going on. Anyways the part of question I've been trying to crack is:

For real numbers a, r and natural number n
let s_n= a + ar + ... arⁿ Show that rs_n– s_n= a(rⁿ⁺¹-1)
And deduce that
| s_n–

\frac{a}{1-r}

| = |

\frac{r ^{n+1}}{1-r}

|
For |r| < 1 deduce that s_n -> a/(1-r) as n -> infinity.

The rs_n– snbit it easy enough. The bit after that I got |

\frac{ar ^{n+1}}{1-r}

| and I’m not sure where I made a mistake but I’m sure its minor so I’m not too bothered about that. But the last bit I know that as n approaches infinity arⁿ will approach 0. Also if r = 0 its easy to see from the sequence and formulae we are showing that the series will equal a. However I don't know where to go from here. Can someone please point out where to go next, I would really appreciate that. Although I need help with 2 parts of this question it’s really the later part that’s bothering me cause I can’t work out infinite series L.