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Show that, if $c$ is a positive real number, then $g(n)=1+c+c^{2}+\dots+c^{n}$ is: (a) $\Theta(1)$ if $c<1$ (b) $\Theta(n)$ if $c=1$ (c) $\Theta\left(c^{n}\right...

Question

Show that, if $c$ is a positive real number, then $g(n)=1+c+c^{2}+\dots+c^{n}$ is: (a) $\Theta(1)$ if $c<1$ (b) $\Theta(n)$ if $c=1$ (c) $\Theta\left(c^{n}\right)$ if $c>1$ The moral: in big- - $\Theta$ terms, the sum of a geometric series is simply the first term if the series is strictly decreasing, the last term if the series is strictly increasing, or the number of terms if the series is unchanging.

Show that, if $c$ is a positive real number, then $g(n)=1+c+c^{2}+\dots+c^{n}$ is: (a) $\Theta(1)$ if $c<1$ (b) $\Theta(n)$ if $c=1$ (c) $\Theta\left(c^{n}\right)$ if $c>1$ The moral: in big- - $\Theta$ terms, the sum of a geometric series is simply the first term if the series is strictly decreasing, the last term if the series is strictly increasing, or the number of terms if the series is unchanging.



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Consider the series $ \sum_{n = 1}^{\infty} n/(n + 1)!. $
(a) Find the partial sums $ s_1, s_2, s_3, $ and $ s_4. $ Do you recognize the denominators? Use the pattern to guess a formula for $ s_n. $
(b) Use mathematical induction to prove your guess.
(c) Show that the given infinite series is convergent, and find its sum.


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