Let `a_1, a_2, a_3,…,a_n`, be a given sequence. Then, the expression
`a_1 + a_2 + a_3 +,…+ a_n + ...`
is called the series associated with the given sequence.
Series are often represented in compact form, called sigma notation, using the Greek letter ∑(sigma) as means of indicating the summation involved.
Thus the series `a_1 + a_2 + a_3 +...+a_n` is abbreviated as `sum _(k=1)^n a_k.`
Find the sum of the following series up to n terms `1^3/1 + (1^3 + 2^3)/(1+3) + (1^3 + 2^3 + 3^3)/(1 + 3 + 5) +...`
Write the first five terms of the following sequence and obtain the corresponding series:
`a_1 = a_2 = 2, a_n = a_(n-1) -1, n > 2`
Write the first five terms of the following sequence and obtain the corresponding series: `a_1 = -1, a_n = (a_(n-1))/n , n >= 2`