Determine whether the statement is true or false. f(x)=(1)/(1-x)=sum_(n=0)^(oo)x^(n) and g(x)=(1)/(1-x^(2))=sum_(n=0)^(oo)x^(2n), then f(x)g(x)=sum_(n=0)^(oo)x^(n)x^(2n)=sum_(n=0)^(oo)x^(3n) True False

Evaluate f(x): Evaluate f(x) = (1)/(1-x) as a power series. f(x) = sum_(n=0)^(oo)x^(n)Evaluate g(x): Evaluate g(x) = (1)/(1-x^(2)) as a power series. g(x) = sum_(n=0)^(oo)x^(2n)Multiply f(x) and g(x): Multiply f(x) and g(x) to find f(x)g(x). f(x)g(x) = (sum_(n=0)^(oo)x^(n))(sum_(m=0)^(oo)x^(2m))Expand the product: Expand the product of the two series. f(x)g(x) = sum_(n=0)^(oo) sum_(m=0)^(oo) x^(n+2m)Simplify the expression: Simplify the expression for the product. f(x)g(x) = sum_(k=0)^(oo) x^(k) where k = n + 2m, k takes values 0, 1, 2, 3, ...

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Calculus

Find derivatives of using multiple formulae

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Q. Determine whether the statement is true or false.

\newline

f(x)=\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}

and

g(x)=\frac{1}{1-x^{2}}=\sum_{n=0}^{\infty} x^{2 n}

, then

f(x) g(x)=\sum_{n=0}^{\infty} x^{n} x^{2 n}=\sum_{n=0}^{\infty} x^{3 n}

\newline

True

\newline

False

Evaluate $f(x)$ : Evaluate $f(x) = \frac{1}{1-x}$ as a power series. $\newline$ $f(x) = \sum_{n=0}^{\infty}x^{n}$
Evaluate $g(x)$ : Evaluate $g(x) = \frac{1}{1-x^{2}}$ as a power series. $\newline$ $g(x) = \sum_{n=0}^{\infty}x^{2n}$
Multiply $f(x)$ and $g(x)$ : Multiply $f(x)$ and $g(x)$ to find $f(x)g(x)$ . $\newline$ f(x)g(x) = \left(\sum_{n=$0$}^{\infty}x^{n}\right)\left(\sum_{m=$0$}^{\infty}x^{$2$m}\right)
Expand the product: Expand the product of the two series. \(f(x)g(x) = \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} x^{n+2m}
Simplify the expression: Simplify the expression for the product. $\newline$ $f(x)g(x) = \sum_{k=0}^{\infty} x^{k}$ where $k = n + 2m$ , $k$ takes values $0, 1, 2, 3, \ldots$