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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.10.29
Capitolo 10, Problema 10.10.29

Use series to evaluate the limits in Exercises 29–40.
29. lim (x → 0) (e^x - (1 + x)) / x²

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Recall the Maclaurin series expansion for the exponential function \(e^x\), which is given by: \[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\]
Substitute the series expansion of \(e^x\) into the given limit expression: \[\lim_{x \to 0} \frac{e^x - (1 + x)}{x^2} = \lim_{x \to 0} \frac{\left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\right) - (1 + x)}{x^2}\]
Simplify the numerator by canceling out the \(1\) and \(x\) terms: \[\lim_{x \to 0} \frac{\frac{x^2}{2!} + \frac{x^3}{3!} + \cdots}{x^2}\]
Divide each term in the numerator by \(x^2\): \[\lim_{x \to 0} \left( \frac{1}{2!} + \frac{x}{3!} + \frac{x^2}{4!} + \cdots \right)\]
Evaluate the limit as \(x\) approaches 0 by substituting \(x = 0\) into the simplified expression, which leaves only the constant term(s).

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Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. For functions like e^x, the series expansion around x=0 (Maclaurin series) helps approximate the function using polynomials, which simplifies limit evaluation.
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Limit Evaluation Using Series

By substituting the series expansion into the limit expression, complex functions can be approximated by polynomials. This allows direct simplification and evaluation of limits, especially when the original expression results in indeterminate forms like 0/0.
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Handling Indeterminate Forms

Limits that yield forms like 0/0 require algebraic manipulation or series expansion to resolve. Using series expansions helps rewrite the numerator and denominator to identify the leading terms, enabling the determination of the limit's value.
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Pratica correlata
Domanda del libro di testo

Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.

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Finding Taylor Polynomials

In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.

f(x) = sin x,a = 0

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Finding Taylor Series

Use substitution (as in Formula (7)) to find the Taylor series at x = 0 of the functions in Exercises 1–12.

e⁻ˣ/²

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Determining Convergence or Divergence

Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

∑ (from n=1 to ∞) eⁿ / (1 + e²ⁿ)

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Finding Terms of a Sequence

Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.

aₙ = 1 / n!

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Domanda del libro di testo

In Exercises 15–22, determine if the geometric series converges or diverges. If a series converges, find its sum.

1 − (2/e) + (2/e)² − (2/e)³ + (2/e)⁴ − …

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