Consider the following series. [infinity] 1 n4 n = 1 (a) Use the sum of the first 10 terms to estimate the sum of the given series. (Round the answer to six decimal places.) s10 = (b) Improve this estimate using the following inequalities with n = 10. (Round your answers to six decimal places.) sn + [infinity] f(x) dx n + 1 ≤ s ≤ sn + [infinity] f(x) dx n ≤ s ≤ (c) Using the Remainder Estimate for the Integral Test, find a value of n that will ensure that the error in the approximation s ≈ sn is less than 0.00001. n > 32 n > −32 n > 5 n > 0 n > 15

Inequality equation are those algebraic equations in which the the two equations compare with greater than sign less than sign or other inequality sign.The value of n for the given problem is,

[tex]n>3[/tex]

Given-

The given series is,

[tex]\sum_{n=1}^{\infty}\dfrac{1}{n^4}[/tex]

Using the value of n equal to 1,2,3..... we can find the sum of ten terms with the help of above formula. Therefore,

[tex]\sum_{n=1}^{10}\dfrac{1}{n^4}=1+\dfrac{1}{2^4} +\dfrac{1}{3^4} +\dfrac{1}{4^4} +\dfrac{1}{5^4} +\dfrac{1}{6^4} +\dfrac{1}{7^4} +\dfrac{1}{8^4} +\dfrac{1}{9^4} +\dfrac{1}{10^4}[/tex]

[tex]\sum_{n=1}^{10}\dfrac{1}{n^4}=1.08204[/tex]

Inequality equation

Inequality equation are those algebraic equations in which the the two equations compare with greater than sign less than sign or other inequality sign.

Now let function,

[tex]f(x) =\dfrac{1}{x^4}[/tex]

By the inequality, we can use the formula,

[tex]s_n+\int\limits^\infty_{n=1} {f(x)} \, dx \leq s \leq s_n+\int\limits^\infty_{n} {f(x)} \, dx[/tex]

For n=10, we get,

[tex]1.08204+\int\limits^\infty_{11} {\dfrac{1}{x^4} } \ dx \leq s\leq 1.08204+\int\limits^\infty_{10} {\dfrac{1}{x^4} } \, dx[/tex]

on solving the above we get,

[tex]1.08228\leq s \leq 1.08237[/tex]

Now by the definition of the function we can write that,

[tex]\int\limits^\infty_{n=1} {f(x)} \, dx \leq s -s_n \leq \int\limits^\infty_{n} {f(x)} \, dx[/tex]

[tex]\int\limits^\infty_{n+1} {\dfrac{1}{x^4}} \, dx \leq s -s_n \leq \int\limits^\infty_{n+1} {\dfrac{1}{x^4} } \, dx[/tex]

on solving it, we get,

[tex]\left ( \dfrac{1}{(n+1)^3} \right ) \leq 0.00001\leq \dfrac{1}{3}\times \dfrac{1}{n^3}[/tex]

on solving the above equation we get,

[tex]n>3[/tex]

Thus the value of n for the given problem is,

[tex]n>3[/tex]

For more about the inequality equation follow the link below-

https://brainly.com/question/11897796