∑ n = 0 ∞ a n ( x − c ) n = a 0 + a 1 ( x − c ) + a 2 ( x − c ) 2 + . . . + a n ( x − c ) n {\displaystyle \sum _{n=0}^{\infty }a_{n}(x-c)^{n}=a_{0}+a_{1}(x-c)+a_{2}(x-c)^{2}+...+a_{n}(x-c)^{n}}
sırasına c {\displaystyle c} nöqtəsində qüvvət sırası deyilir. Burada a n {\displaystyle a_{n}} əmsalları ədədlərdir. Xüsusi halda c = 0 {\displaystyle c=0} olarsa, onda ∑ n = 0 ∞ a n x n = a 0 + a 1 x + a 2 x 2 + ⋯ . {\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots .} Bu sıraya sıfır nöqtəsində qüvvət sırası deyilir.
e x = ∑ n = 0 ∞ x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + ⋯ , {\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots ,}
sin ( x ) = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ , {\displaystyle \sin(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots ,}
f ( x ) = ∑ n = 0 ∞ a n ( x − c ) n {\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}(x-c)^{n}}
g ( x ) = ∑ n = 0 ∞ b n ( x − c ) n {\displaystyle g(x)=\sum _{n=0}^{\infty }b_{n}(x-c)^{n}}
onda
f ( x ) ± g ( x ) = ∑ n = 0 ∞ ( a n ± b n ) ( x − c ) n . {\displaystyle f(x)\pm g(x)=\sum _{n=0}^{\infty }(a_{n}\pm b_{n})(x-c)^{n}.}
f ( x ) g ( x ) = ( ∑ n = 0 ∞ a n ( x − c ) n ) ( ∑ n = 0 ∞ b n ( x − c ) n ) {\displaystyle f(x)g(x)=\left(\sum _{n=0}^{\infty }a_{n}(x-c)^{n}\right)\left(\sum _{n=0}^{\infty }b_{n}(x-c)^{n}\right)}
= ∑ i = 0 ∞ ∑ j = 0 ∞ a i b j ( x − c ) i + j {\displaystyle =\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }a_{i}b_{j}(x-c)^{i+j}}
= ∑ n = 0 ∞ ( ∑ i = 0 n a i b n − i ) ( x − c ) n . {\displaystyle =\sum _{n=0}^{\infty }\left(\sum _{i=0}^{n}a_{i}b_{n-i}\right)(x-c)^{n}.}
f ( x ) g ( x ) = ∑ n = 0 ∞ a n ( x − c ) n ∑ n = 0 ∞ b n ( x − c ) n = ∑ n = 0 ∞ d n ( x − c ) n {\displaystyle {f(x) \over g(x)}={\sum _{n=0}^{\infty }a_{n}(x-c)^{n} \over \sum _{n=0}^{\infty }b_{n}(x-c)^{n}}=\sum _{n=0}^{\infty }d_{n}(x-c)^{n}}
f ′ ( x ) = ∑ n = 1 ∞ a n n ( x − c ) n − 1 = ∑ n = 0 ∞ a n + 1 ( n + 1 ) ( x − c ) n {\displaystyle f^{\prime }(x)=\sum _{n=1}^{\infty }a_{n}n\left(x-c\right)^{n-1}=\sum _{n=0}^{\infty }a_{n+1}\left(n+1\right)\left(x-c\right)^{n}}
∫ f ( x ) d x = ∑ n = 0 ∞ a n ( x − c ) n + 1 n + 1 + k = ∑ n = 1 ∞ a n − 1 ( x − c ) n n + k . {\displaystyle \int f(x)\,dx=\sum _{n=0}^{\infty }{\frac {a_{n}\left(x-c\right)^{n+1}}{n+1}}+k=\sum _{n=1}^{\infty }{\frac {a_{n-1}\left(x-c\right)^{n}}{n}}+k.}
