Chapter 7: Problem 114
In Exercises 113 and \(114,\) find the values of \(p\) for which the series converges. $$ \sum_{n=1}^{\infty}(-1)^{n}\left(\frac{1}{n+p}\right) $$
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Compute the first six terms of the sequence \(\left\\{a_{n}\right\\}=\\{\sqrt[n]{n}\\} .\) If the sequence converges, find its limit.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The series \(\sum_{n=1}^{\infty} \frac{n}{1000(n+1)}\) diverges.
The Fibonacci sequence is defined recursively by \(a_{n+2}=a_{n}+a_{n+1},\) where \(a_{1}=1\) and \(a_{2}=1\) (a) Show that \(\frac{1}{a_{n+1} a_{n+3}}=\frac{1}{a_{n+1} a_{n+2}}-\frac{1}{a_{n+2} a_{n+3}}\). (b) Show that \(\sum_{n=0}^{\infty} \frac{1}{a_{n+1} a_{n+3}}=1\).
In Exercises \(103-106,\) use the formula for the \(n\) th partial sum of a geometric series $$\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}$$ The winner of a \(\$ 1,000,000\) sweepstakes will be paid \(\$ 50,000\) per year for 20 years. The money earns \(6 \%\) interest per year. The present value of the winnings is \(\sum_{n=1}^{20} 50,000\left(\frac{1}{1.06}\right)^{n}\) Compute the present value and interpret its meaning.
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{3 n-1}{2 n+1} $$
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