Domanda del libro di testo
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = 1/x², a=1
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 11, Problema 11.3.23c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = cosh 3x, a = 0
Guida verificata passo dopo passo1
Recall that the Taylor series expansion of \(f(x) = \cosh(3x)\) about \(a = 0\) can be found by using the known series for \(\cosh x\), which is \(\cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}\). Replace \(x\) by \$3x$ to write the series for \(\cosh(3x)\) as \(\sum_{n=0}^{\infty} \frac{(3x)^{2n}}{(2n)!}\).
Express the series explicitly as \(\sum_{n=0}^{\infty} \frac{3^{2n} x^{2n}}{(2n)!}\). This is the Taylor series centered at \(a=0\) for \(f(x)\).
To determine the interval of convergence, apply the Ratio Test to the general term \(a_n = \frac{3^{2n} x^{2n}}{(2n)!}\). Compute the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
Simplify the ratio inside the limit: \(\left| \frac{3^{2(n+1)} x^{2(n+1)}}{(2(n+1))!} \cdot \frac{(2n)!}{3^{2n} x^{2n}} \right| = \left| \frac{3^{2} x^{2}}{(2n+2)(2n+1)} \right|\). Then evaluate the limit as \(n \to \infty\).
Since the factorial in the denominator grows faster than any polynomial, the limit \(L\) approaches zero for all real \(x\). Therefore, the series converges for all real numbers, and the interval of convergence is \((-\infty, \infty)\).
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Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point. For f(x) centered at a, the series is ∑ (f⁽ⁿ⁾(a)/n!) (x - a)ⁿ. It approximates functions like cosh(3x) using polynomial terms around a = 0.
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Taylor Series
Interval of Convergence
The interval of convergence is the set of x-values for which the Taylor series converges to the function. It is found by applying convergence tests (like the ratio test) to the series terms, determining where the infinite sum is valid and accurately represents the function.
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08:44Interval of Convergence
Hyperbolic Cosine Function (cosh x)
The hyperbolic cosine function, cosh x, is defined as (eˣ + e⁻ˣ)/2 and is an even, smooth function. Its Taylor series at 0 includes only even powers of x with positive coefficients, which helps in forming the series for cosh(3x) by substituting 3x into the expansion.
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Percorso guidato
5:53Graph of Sine and Cosine Function
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Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = e⁻ˣ, a=0
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