Write the following series using sigma notation:
(x/3)-((4x^2)/9)+(x^3/3)-((16x^4)/81)+((25x^5)/243)-...
To write the given series using sigma notation, we need to identify the pattern of the terms and determine the indices.
Looking at the terms, we can observe that the coefficient of each term alternates between positive and negative values. Also, the exponent of "x" in the numerators follows the pattern of an odd power of "x" which matches the cube power, i.e., x, x^3, x^5, etc.
To represent this pattern using sigma notation, let's define the general term of the series. The general term can be expressed as follows:
a(n) = (-1)^(n+1) * (n^2 * x^n) / (3^n)
Here, a(n) represents the nth term of the series.
Now we can write the series using sigma notation:
∑ [(-1)^(n+1) * (n^2 * x^n) / (3^n)] from n = 1 to ∞
In this sigma notation, the variable "n" represents the index of the terms, which starts from 1 and goes to infinity (∞).
Please note that the series is an infinite series as there's no specified number of terms mentioned. The (n^2) in the numerator represents the square of the index because it starts from n = 1.
So, the sigma notation for the given series is:
∑ [(-1)^(n+1) * (n^2 * x^n) / (3^n)] from n = 1 to ∞