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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 10 - Sequences and Infinite Series문제 10.7.37
10장, 문제 10.7.37

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) k¹⁰⁰ / (k + 1)!

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{k^{100}}{(k+1)!} \). We want to determine if it converges absolutely, conditionally, or diverges.
Since the terms involve factorials, consider using the Ratio Test, which is effective for series with factorial expressions.
Set up the Ratio Test by examining the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \), where \( a_k = \frac{k^{100}}{(k+1)!} \). Write \( a_{k+1} = \frac{(k+1)^{100}}{(k+2)!} \).
Compute the ratio \( \frac{a_{k+1}}{a_k} = \frac{(k+1)^{100}}{(k+2)!} \times \frac{(k+1)!}{k^{100}} \). Simplify the factorial terms and the powers to express the ratio in a form suitable for taking the limit as \( k \to \infty \).
Evaluate the limit \( L \). If \( L < 1 \), the series converges absolutely; if \( L > 1 \), it diverges; if \( L = 1 \), the test is inconclusive and another test must be used.

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주요 개념

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Absolute and Conditional Convergence

A series converges absolutely if the series of absolute values converges. If the original series converges but not absolutely, it converges conditionally. Understanding these distinctions helps classify the behavior of infinite series.
추천 영상:
가이드 코스
07:51
Choosing a Convergence Test

Ratio Test

The Ratio Test determines convergence by examining the limit of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges. It is especially useful for series involving factorials.
추천 영상:
가이드 코스
05:35
Ratio Test

Factorials and Growth Rates

Factorials grow faster than any polynomial function. Recognizing how factorial terms dominate polynomial terms in the numerator or denominator helps predict the behavior of series and guides the choice of convergence tests.
추천 영상:
5:22
Factorials
관련 실천
교과서 질문

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁. 

aₙ = 1/10ⁿ

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교과서 질문

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


29.∑ (k = 1 to ∞) e^(–2k)

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교과서 질문

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 0 to ∞) 1 / (1000 + k)

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교과서 질문

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 1 to ∞) (2 + (−1)ᵏ) / k²

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교과서 질문

16–17. {Use of Tech} Periodic savings

Suppose you deposit m dollars at the beginning of every month in a savings account that earns a monthly interest rate of r, which is the annual interest rate divided by 12 (for example, if the annual interest rate is 2.4%, r = 0.024/12 = 0.002). For an initial investment of m dollars, the amount of money in your account at the beginning of the second month is the sum of your second deposit and your initial deposit plus interest, or m + m(1 + r). Continuing in this fashion, it can be shown that the amount of money in your account after n months is


Aₙ = m + m(1 + r) + ⋯ + m(1 + r)ⁿ⁻¹.


Use geometric sums to determine the amount of money in your savings account after 5 years (60 months) using the given monthly deposit and interest rate.


17. Monthly deposits of \$250 at a monthly interest rate of 0.2%

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교과서 질문

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

{Use of Tech} ∑ (k = 1 to ∞) 1 / (4lnk)

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