Chapter 8: Series

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?

\(\sum\limits_{n = 1}^\infty {\frac{1}{{{n^3}}}} \)

Find the Maclaurin series for \(f\) and its radius of the convergence.

The \(n\)th derivative of the function \(f(x)\) at the point 0 is, \({f^n}(0) = (n + 1)!\) for\(n = 1,2,3, \cdots \).

Determine the radius of convergence and interval of convergence of a series, \(\sum\limits_{n = 1}^\infty {{{( - 1)}^n}} n{x^n}\).

Determine whether the sequence is convergent or divergent. If it is convergent, find its limit\({{\rm{a}}_{\rm{n}}}{\rm{ = }}\frac{{{{\rm{n}}^{\rm{3}}}}}{{{\rm{1 + }}{{\rm{n}}^{\rm{2}}}}}\).

(a) Determine the sum of the series, starting with the geometric series\(\sum\limits_{n = 0}^\infty {{x^n}} \).

(b) (i) Determine the sum of the series\(\sum\limits_{n = 1}^\infty n {x^n},|x| < 1\).

(ii) Determine the sum of the series\(\sum\limits_{n = 1}^\infty {\frac{n}{{{2^n}}}} \).

(c) (i) Determine the sum of the series\(\sum\limits_{n = 2}^\infty n (n - 1){x^n},|x| < 1\).

(ii) Determine the sum of the series\(\sum\limits_{n = 2}^\infty {\frac{{{n^2} - n}}{{{2^n}}}} \).

(iii) Determine the sum of the series\(\sum\limits_{n = 1}^\infty {\frac{{{n^2}}}{{{2^n}}}} \).

\({\bf{37 - 40}}\) Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

\({a_n} = n + \frac{1}{n}\)

If the nth partial sum of a series \(\sum\limits_{n = 1}^\infty {{a_n}} \) is \({s_n} = 3 - n{2^{ - n}}\), find \({a_n}\)and \(\sum\limits_{n = 1}^\infty {{a_n}} \).

Use the Maclaurin series for \({e^x}\) to compute \(1/\sqrt({10}){e}\) correct

to five decimal places.

Find the radius of convergence of the series \(\sum\limits_{{\rm{n = 1}}}^\infty {\frac{{{\rm{(2n)!}}}}{{{{{\rm{(n!)}}}^{\rm{2}}}}}} {{\rm{x}}^{\rm{n}}}\)

A patient takes 150 mg of a drug at the same time every day. Just before each tablet is taken, 5% of the drug remains in the body.
(a) What quantity of the drug is in the body after the third tablet? After the n th tablet?
(b) What quantity of the drug remains in the body in the long run?