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  • <img alt="\mathrm{\lim _{n \rightarrow \infty}\left[\sum_{r=1}^n \frac{1}{2^r}\right]}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%7Bn%20%5Crightarrow%20%5Cinfty%7D

\mathrm{\lim _{n \rightarrow \infty}\left[\sum_{r=1}^n \frac{1}{2^r}\right]}, where \mathrm{\left [ . \right ]} denotes the greatest integer function, is

Option: 1

equal to one


Option: 2

 equal to zero


Option: 3

non-existent


Option: 4

none of these


Answers (1)

best_answer

\mathrm{\sum_{r=1}^n \frac{1}{2^r}=\frac{\frac{1}{2}\left(1-\left(\frac{1}{2}\right)^n\right)}{\left(1-\frac{1}{2}\right)}=1-\left(\frac{1}{2}\right)^n}, which tends to one as \mathrm{n \rightarrow \infty} (but infact always remains less than one).
Thus \mathrm{\lim _{n \rightarrow \infty}\left[\sum_{r=1}^n \frac{1}{2^r}\right]=0}.

Posted by

Pankaj

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