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Section 10.15 : Power Series and Functions
4. Give a power series representation for the derivative of the following function.
\[g\left( x \right) = \frac{{5x}}{{1 - 3{x^5}}}\]Show All Steps Hide All Steps
Hint : While we could differentiate the function and then attempt to find a power series representation that seems like a lot of work. It’s a good think that we know how to differentiate power series.
First let’s notice that we can quickly find a power series representation for this function. Here is that work.
\[g\left( x \right) = 5x\frac{1}{{1 - 3{x^5}}} = 5x\sum\limits_{n = 0}^\infty {{{\left( {3{x^5}} \right)}^n}} = \sum\limits_{n = 0}^\infty {5x\,\left( {{3^n}} \right){x^{5n}}} = \sum\limits_{n = 0}^\infty {5\left( {{3^n}} \right){x^{5n + 1}}} \] Show Step 2Now, we know how to differentiate power series and we know that the derivative of the power series representation of a function is the power series representation of the derivative of the function.
Therefore,
\[g'\left( x \right) = \frac{d}{{dx}}\left[ {\sum\limits_{n = 0}^\infty {5\left( {{3^n}} \right){x^{5n + 1}}} } \right] = \require{bbox} \bbox[2pt,border:1px solid black]{{\sum\limits_{n = 0}^\infty {5\left( {5n + 1} \right)\left( {{3^n}} \right){x^{5n}}} }}\]Remember that to differentiate a power series all we need to do is differentiate the term of the power series with respect to \(x\).