If x>1 find the sum of infinity of x ^2/(x-1)+x+(x-1).... And also find the value of x for the given series TO HAVE THE SUM OF INFINITY
To find the sum of an infinite series, we need to analyze the convergence of the series. In this case, we have to check whether the series converges or diverges.
The given series is:
S = x^2/(x-1) + x + (x-1) + ...
To determine convergence, let's rearrange the terms and write the series in a different form. We can rewrite the above series as:
S = x^2/(x-1) + x + x - 1 + x + 1 - 1 + ...
Simplifying this expression, we get:
S = (x^2 + 2x)/(x-1) + (x + x + x + ...)
Now, let's examine each part of the series separately:
1. (x^2 + 2x)/(x-1) = x^2/(x-1) + 2x/(x-1)
The first term, x^2/(x-1), can be simplified using polynomial division or partial fraction decomposition:
x^2/(x-1) = x + 1 + 1/(x-1)
So, x^2/(x-1) can be written as x + 1 + 1/(x-1).
2. (x + x + x + ...) = x + x + x + ...
This part is an infinite geometric series with a common ratio of 1. The sum of this geometric series can be found using the formula S = a/(1 - r), where 'a' is the first term and 'r' is the common ratio. In this case, a = x and r = 1.
So, using the formula for the sum of an infinite geometric series, we have:
(x + x + x + ...) = x/(1 - 1) = undefined
Now, let's combine the two parts of the series:
S = (x^2 + 2x)/(x-1) + (x + x + x + ...) = x + 1 + 1/(x-1) + undefined
From the above expression, we can see that the series is undefined and does not converge. Therefore, the sum of an infinite number of terms for this series does not exist for any value of x, including x > 1.
In conclusion, the series does not have a sum of infinity for any value of x, as it is divergent.