Question

89–90. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, complete the following.

c. Find lower and upper bounds (Lₙ and Uₙ respectively) for the exact value of the series.

89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

Accepted Answer

Identify the series given: $$ \sum_{k=1}^{\infty} \frac{1}{k^5} . $$This is a p-series with $$ p = 5 $$, which converges because $$ p \u003e 1 . $$Recognize that the partial sum $$ S_n = \sum_{k=1}^n \frac{1}{k^5} $$ approximates the total sum, and the remainder (or tail) $$ R_n = \sum_{k=n+1}^\infty \frac{1}{k^5} $$ represents the error between $$ S_n $$ and the exact sum. Use the integral test remainder estimates to find bounds for the remainder $$ R_n . $$The integral test tells us that for a decreasing positive function $$ f(k) = \frac{1}{k^5} $$, the remainder satisfies:

$$ \int_{n+1}^\infty f(x) \, dx \leq R_n \leq \int_n^\infty f(x) \, dx $$

which translates to:

$$ \int_{n+1}^\infty \frac{1}{x^5} \, dx \leq R_n \leq \int_n^\infty \frac{1}{x^5} \, dx $$ Calculate these improper integrals (without evaluating the final numeric value) using the formula for integrals of power functions:

$$ \int_a^\infty x^{-p} \, dx = \frac{a^{-(p-1)}}{p-1} \quad 	ext{for} \quad p \u003e 1 $$

Apply this to $$ p = 5 $$ and $$ a = n $$ and $$ a = n+1  to $$express the bounds for $$ R_n . $$Finally, express the lower and upper bounds for the exact sum of the series as:

$$ L_n = S_n + \int_{n+1}^\infty \frac{1}{x^5} \, dx \quad 	ext{and} \quad U_n = S_n + \int_n^\infty \frac{1}{x^5} \, dx $$

where $$ S_n = \sum_{k=1}^n \frac{1}{k^5}  is $$the partial sum up to $$ n = 5 .$$

89–90. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, complete the following.

c. Find lower and upper bounds (Lₙ and Uₙ respectively) for the exact value of the series.

89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

Verified video answer for a similar problem:

Key Concepts

Convergent Infinite Series

Partial Sums and Remainder Estimation

Integral Test for Bounding Series Remainders

89–90. {Use of Tech} Lower and upper bounds of a seriesFor each convergent series and given value of n, complete the following.c. Find lower and upper bounds (Lₙ and Uₙ respectively) for the exact value of the series.89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

Verified video answer for a similar problem:

Key Concepts

Convergent Infinite Series

Partial Sums and Remainder Estimation

Integral Test for Bounding Series Remainders

89–90. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, complete the following.

c. Find lower and upper bounds (Lₙ and Uₙ respectively) for the exact value of the series.

89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5