Property 3
Series
\((a_n)\) (a sequence)
\(\sum\limits_{n=1}^{\infty}a_n=a_1+a_2+\cdots+a_n+\cdots\)
(the sum is called series.)
(To find the result of the series;)
(1.series of partial sum is found.)
(2. the limit of the defined series is taken)
- \(S_n=\sum\limits_{k=1}^n a_k\) Series of partial sum
- \(\sum\limits_{n=1}^{\infty}a_n=\lim\limits_{n\to\infty}(S_n)\)
Note
The result of the series to be a real number, the limit of the series should definetely be 0. However, if the limit of the series is 0 the result of the series doesn't require to be a real number.
\((a_n)\) (one geometrical series)
\((a_n)=(a_1\cdot r^{n-1})\)
\(\sum\limits_{k=1}^{\infty}a_k=\samefrac{a_1}{1-r}\qquad |r|<1\)
1.
\(\sum\limits_{k=1}^{\infty}\samefrac1{(k+1)\cdot(k+2)}=?\)
2.
\(\sum\limits_{k=3}^{\infty}\samefrac1{k\cdot(k+1)}=?\)
3.
\(\sum\limits_{n=3}^{\infty}\samefrac1{n\cdot(n-2)}=?\)
4.
\(\sum\limits_{n=1}^{\infty}\left(\samefrac14\right)^{n+2}=?\)
5.
\(\sum\limits_{k=0}^{\infty}2^{1-k}=?\)
6.
\(1+\samefrac25+\samefrac4{25}+\cdots+\left(\samefrac25\right)^n+\cdots=?\)
7.
\(\sum\limits_{n=1}^{\infty}(-1)^n\cdot\left(\samefrac23\right)^n=?\)