Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = cosh 3x, a = 0

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Recall the definition of the hyperbolic cosine function: $\cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}$. This is the Maclaurin series expansion (Taylor series at $a=0$) for $\cosh x$.

Since the function is $f(x) = \cosh 3x$, substitute \$3x$ in place of $x$ in the series expansion. This gives $f(x) = \sum_{n=0}^{\infty} \frac{(3x)^{2n}}{(2n)!}$.

Simplify the power inside the summation: $(3x)^{2n} = 3^{2n} x^{2n}$. So the series becomes $f(x) = \sum_{n=0}^{\infty} \frac{3^{2n} x^{2n}}{(2n)!}$.

Express the power series explicitly in summation notation centered at $a=0$ as $f(x) = \sum_{n=0}^{\infty} \frac{3^{2n}}{(2n)!} (x - 0)^{2n}$.

This is the power series representation of $\cosh 3x$ about $a=0$ in summation notation.

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주요 개념

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Taylor Series

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. For a function f(x) centered at a = 0, it is expressed as the sum of (f^n(0)/n!) * x^n, where f^n(0) is the nth derivative evaluated at zero. This series approximates the function near the center point.

Power Series and Summation Notation

A power series is an infinite series of the form Σ c_n (x - a)^n, where c_n are coefficients and a is the center. Summation notation compactly expresses this infinite sum using the sigma symbol (Σ), indicating the sum over all terms indexed by n. Writing a function as a power series helps analyze and approximate it.

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges to a finite value. It is found by applying convergence tests like the ratio test. Understanding this interval is crucial because the Taylor series only accurately represents the function within this range.

Taylor series and interval of convergenceb. Write the power series using summation notation.f(x) = cosh 3x, a = 0