Question

state and prove cauchy's general principle of convergence .

Answer 1

Given:

We have cauchy's general principle of convergence

To Find:

State and prove cauchy's general principle of convergence .

Step-by-step explanation:

Defination:-An infinite series converges iff for every ε > 0, there exists a positive integer N such that

$\begin{vmatrix}x_n_+_1+\cdots+x_n\end{vmatrix}< \textrm{e whwnever m} \geq n\geq N$

Proof:

• Let we will start with

         $S_m=(x_1+x_2+\cdots+x_m)and S_n=(x_1+x_2+\cdots+x_n)$

         be the mth and nth partial sum of the series, where m is greater then  n

                this implies

              =│$S_m-S_n$│

              =│$(x_1+x_2+\cdots+x_m)-(x_1+x_2+\cdots+x_n)$)│

              =│$(x_n_+_1+\cdots+x_m)$│

•  Now for e > 0 given,the series

• converges iff sequence of partial sums {Sn} converges

• Thus │$S_m-S_n$│ < ε for every m≥ n ≥ N for some N∈N

• Hence,│$(x_n_+_1+\cdots+x_m)$│<εfor every m≥ n ≥ N for some N∈N

Answer 2

Cauchy's general principle of convergence: An infinite series converges iff for every ε > 0, there exists a positive integer N such that ││< ε whenever m ≥ n ≥ N. Proof: Let S m = (x 1 + x 2 + ……. + x m) and S n = (x 1 + x 2 + ……….+ x n) be the m th and n th partial sum of the series, where m ≥ n. │S m – S n │ =│(x 1 + x 2 + ….. + x n + x n+1 +……. + x m)-(x 1 + x 2 + ……….+ x n)│ =│x n+1 +……. + x m │ Now for ε > 0 given, the series converges iff sequence of partial sums {Sn} converges │S m – S n │ < ε for every m ≥ n ≥ N for some N ∈ │x n+1 +……. + x m │< ε for every m ≥ n ≥ N for some N ∈. Proved.