![JC16](/avatars/33134.jpg)
JC16
20.08.2019 •
Mathematics
Shannon rolls a number cube and then chooses a card from set of cards numbered 1 through 5. what is the proability that she will roll an even number and choose an odd- numbered card?
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Ответ:
Ответ:
Step-by-step explanation:
Given series :![[\dfrac{5}{1\cdot2}]+[\dfrac{5}{2\cdot3}]+[\dfrac{5}{3\cdot4}]+....+[\dfrac{5}{n\cdot(n+1)}]](/tpl/images/0387/4588/7ac51.png)
Sum of series =![S_n=\sum^{\infty}_{1}\ [\dfrac{5}{n\cdot(n+1)}]=5[\sum^{\infty}_{1}\dfrac{1}{n\cdot(n+1)}]](/tpl/images/0387/4588/bc6b8.png)
Consider![\dfrac{1}{n\cdot(n+1)}=\dfrac{n+1-n}{n(n+1)}](/tpl/images/0387/4588/6b8ee.png)
⇒![S_n=5\sum^{\infty}_{1}\dfrac{1}{n\cdot(n+1)}=5\sum^{\infty}_{1}[\dfrac{1}{n}-\dfrac{1}{n+1}]](/tpl/images/0387/4588/6ce6d.png)
Put values of n= 1,2,3,4,5,.....n
⇒![S_n=5(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+......-\dfrac{1}{n}+\dfrac{1}{n}-\dfrac{1}{n+1})](/tpl/images/0387/4588/56080.png)
All terms get cancel but First and last terms left behind.
⇒![S_n=5(1-\dfrac{1}{n+1})](/tpl/images/0387/4588/9c98a.png)
Formula for the nth partial sum of the series :
Also,![\lim_{n \to \infty} S_n = 5(1-\dfrac{1}{n+1})](/tpl/images/0387/4588/21682.png)