Infinite Series of Geometric Progression

Please note this post is quite advanced, so don't worry if you find this confusing. Remember that … means to carry on the pattern forever.
If you are unfamiliar with infinite series, then put simply it is the summation of an infinite number of terms, for example:

These terms form a sequence with a particular pattern. The patterns above should be quite obvious.
Infinite series either converge or diverge, if a series converges it approaches a specific value, if it diverges it continually grows or declines (approaches plus or minus infinity). This may seem odd but think about this particular series:

This series converges at a particular value, if you add the terms repeatedly you should see it approaches 1 (gets closer to 1 as you add more and more terms). Which is unlike: 

as this series diverges, as it doesn't approach a specific value. The series gets larger each time by a greater and greater amount, so we say the series approaches infinity (the more terms you add on, you should find the series doesn't get closer and closer to a specific value but rather continually increases).

Sigma Notation

We can represent these infinite series more concisely with proper mathematical notation.

Below, the Z-shape is called a sigma and is the Greek capital for the letter s. It means to add up the terms of a sequence for all integers from the bottom limit to the top limit. You can hopefully see this from the 2 examples below:

note: I didn't put the answer to the first equation, but you should see it is just some number

For a series to be infinite the upper bound has to be infinity. We can either say:

We will apply this notation to show the above convergent series:

remember the sigma is just a different way of representing the expression 

Proof of Convergence for the above Equation

To show the above example equals one:

when I divided x by 2, it was the same as subtracting one half

Geometric Progression

A geometric sequence involves multiplying the last term by the common ratio to find the next term. Examples include:

We can see for the first sequence the common ratio is 2, and for the second it is 3. These can be represented by the formulae:

I have only subtracted one from n to shift the sequence to start at 1-don't worry about it

Deriving the Formula for Convergence for the example below

Now we know about geometric progression, we can formalise the proof above. Suppose we have an infinite series, taking the form below:

k is just some real number

We can expand this out, as we know what the sigma represents:

We can divide both sides by k to get the expression below; the expression is just the above X excluding 1/k. So:

We can now solve:

The formula for the value the infinite series converges to as a function of some real number k:

(when k>1, else when k=1 or is smaller than 1 the formula gives illogical results)

When k=2, X is one, which is what we saw above

When k=3, X is one half, which is one third plus one ninth plus one twenty-seventh...

When k=4, X is on third, which is one fourth plus one sixteenth...

I hope you have enjoyed reading this, I feel this may be quite difficult for some, but after all it is a hard topic.