Home » Blog » Determine the interval of convergence of ∑ x2n / n! and show it satisfies a given differential equation

Determine the interval of convergence of ∑ x2n / n! and show it satisfies a given differential equation

by RoRi

Consider the function f(x) defined by the power series

    \[ f(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{n!}. \]

Determine the interval of convergence for f(x) and show that it satisfies the differential equation

    \[ y' = 2xy. \]

First, to determine the interval of convergence we use the ratio test,

    \begin{align*} \lim_{n \to \infty} \frac{a_{n+1}}{a_n} &= \lim_{n \to \infty} \left( \frac{x^{2n+2}}{(n+1)!} \right) \left( \frac{n!}{x^{2n}} \right) \\[9pt] &= \lim_{n \to \infty} \frac{x^2}{n+1} \\[9pt] &= 0. \end{align*}

Hence, the series converges for all x.

To show that it satisfies the given differential equation, we first take the derivative,

    \begin{align*} && f(x) &= \sum_{n=0}^{\infty} \frac{x^{2n}}{n!} \\[9pt] \implies && f'(x) &= \sum_{n=1}^{\infty} \frac{2n x^{2n-1}}{n!} \\[9pt] &&&= 2 \sum_{n=1}^{\infty} \frac{x^{2n-1}}{(n-1)!}. \end{align*}

Then, it satisfies the given differential equation since

    \begin{align*} 2xy &= 2x \left( \sum_{n=0}^{\infty} \frac{x^{2n}}{n!} \right)\\[9pt] &= 2 \left( \sum_{n=0}^{\infty} \frac{x^{2n+1}}{n!} \right)\\[9pt] &= 2 \sum_{n=1}^{\infty} \frac{x^{2(n-1)+1}}{(n-1)!} \\[9pt] &= 2 \sum_{n=1}^{\infty} \frac{x^{2n-1}}{(n-1)!} \\[9pt] &= y'. \end{align*}

Then, since the given differential equation is a first-order linear differential equation of the form

    \[ y' + P(x)y = 0 \]

with P(x) = -2x we know that the solutions are uniquely determined by the formula

    \[ f(x) = be^{-A(x)}, \qquad \text{where} \qquad A(x) = \int_a^x P(t) \, dt. \]

Since we have the initial condition f(0) = 1 (by plugging in x = 0 to the power series expansion for f(x)) we have a= b= 0 and the unique solution of this differential equation is

    \[ f(x) = e^{-A(x)} = \exp \left( - \int_0^x (-2t) \, dt \right) = e^{x^2}. \]

One comment

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):