Use Theorem Alternating Series remainder to determine the number of terms required to approximate the sum of the series with an error of less than 0.001. sum_{n=1}^inftyfrac{(-1)^{n+1}}{n^5}

Question
Series
Use Theorem Alternating Series remainder to determine the number of terms required to approximate the sum of the series with an error of less than 0.001.
$$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^5}$$

2020-12-31
Given that
$$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^5}$$
We need use theorem alternating series to determine the number of terms requires to approximate the sum of the series with an error of less than 0.001.
Given series is alternate series with $$a_n=\frac{1}{n^5}$$
We know that
$$|R_n|\leq a_{N+1}=\frac{1}{(N+1)^5}$$
for an error less than 0.001N must satisfy the inequality.
$$\frac{1}{(N+1)^5}<0.001$$</span>
$$\frac{1}{0.001}<(N+1)^51000<(N+1)^5$$</span> taking $$\frac{1}{5}$$ root on both sides
$$1000^{\frac{1}{5}} \(3.98 \(2.98 So we will need at least 4 terms to get an error less than 0.001 Relevant Questions asked 2021-01-13 Use Theorem Alternating Series remainder to determine the number of terms required to approximate the sum of the series with an error of less than 0.001. \(\sum_{n=1}^\infty\frac{(-1)^{n+1}}{2n^3-1}$$
Consider the series $$\sum_{n=1}^\infty\frac{(-1)^n}{n^2}$$
a) Show the series converges or diverges using the alternating series test.
b) Approximate the sum using the 4-th partial sum($$S_4$$) of the series.
c) Calculate the maximum error between partial sum($$S_4$$) and the sum of the series using the remainder term portion of the alternating series test.
Consider the following convergent series.
a. Find an upper bound for the remainder in terms of n.
b. Find how many terms are needed to ensure that the remainder is less than $$10^{-3}$$.
c. Find lower and upper bounds (ln and Un, respectively) on the exact value of the series.
$$\sum_{k=1}^\infty\frac{1}{3^k}$$
A 1300-kg car coasts on a horizontal road, with a speed of18m/s. After crossing an unpaved sandy stretch of road 30.0 mlong, its speed decreases to 15m/s. If the sandy portion ofthe road had been only 15.0 m long, would the car's speed havedecreasedby 1.5 m/s, more than 1.5 m/s, or less than 1.5m/s?Explain. Calculate the change in speed in that case.
Write out the first few terms of the series
$$\sum_{n=0}^\infty\frac{(-1)^n}{5^n}$$
What is the​ series' sum?
Use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series.
$$\sum_{n=2}^\infty\frac{(-1)^nn}{n^2-3}$$
$$\sum_{n=1}^\infty\frac{(-1)^n}{n^5}$$
Determine whether the series $$\sum a_n$$ an converges or diverges: Use the Alternating Series Test.
$$\sum_{n=2}^\infty(-1)^n\frac{n}{\ln(n)}$$
$$\sum_{n=0}^\infty\frac{3(-2)^n-5^n}{8^n}$$
$$\sum_{n=1}^\infty\frac{3}{\sqrt n}$$