Question

Find the closed form for the following series: 1 + x2 + x3 + … + xn where x is constant and x > 1

Answer 1

So, the given series is like this:

1 + x2 + x3 + … + xn

Now, we can solve this by following these steps:

Let's add and subtract x to the series as it will not effect the series:

1 + x2 + x3 + … + xn + (x - x)    ---------------->(as x-x is 0 so it will not affect the series)

Now, we can write this as :

x1 + x2 + x3 + ... + xn + (1 - x)

Now, take x common from the series:

So, x(1 + 2 + 3 + 4 + ... + n) + (1 - x)   -------->(as we take positive x to the bracket and negative x along with 1 to the separate part)

Now, as we can see that the series forming inside the bracket is the series of the 'n' natural numbers.

Hence, the sum of 'n' natural numbers if n*(n + 1)/2

So, we can write the series like, x*n*(n + 1)/2 + (1 - x).

So, the closed form for the given series is :  

x*n*(n + 1)/2 + (1 - x).

So, this was the solutions of the problem.

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It really helps :)