Question

Prove that an absolutely convergent series $\sum _{m=1}^{\infty }{a}_{\mathbf{n}}$is convergent. Hint: Put${\mathit{b}}_{\mathbf{n}}\mathbf{=}{\mathit{a}}_{\mathbf{n}}\mathbf{+}\mathbf{\left|}\overline{{\mathbf{a}}_{\mathbf{n}}}\right|$. Then theare nonnegative; we have$\mathbf{\left|}{\mathit{b}}_{\mathbf{n}}\mathbf{\right|}\mathbf{\le }\mathbf{2}\mathbf{\left|}\overline{{\mathbf{a}}_{\mathbf{n}}}\mathbf{\right|}$and${\mathit{a}}_{\mathbf{n}}\mathbf{=}{\mathit{b}}_{\mathbf{n}}\mathbf{-}\mathbf{\left|}\overline{{\mathbf{a}}_{\mathbf{n}}}\mathbf{\right|}$

Short answer

Define a sequence$b_{\mathrm{n}}=a_{\mathrm{n}}+\left|{\mathrm{a}}_{\mathrm{n}}\right|$ so that for every. Now use $\left|b_{\mathrm{n}}\right|\le 2\left|a_n\right|$to get a condition for$\sum _{m=1}^{\infty }{b}_{\mathrm{n}}$ and then make use of the first equation again to finish the proof.

Step-by-step solution

Significance of alternating series

If the terms' absolute values gradually decline to zero, or if$\mathbf{\left|}{\mathit{a}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{\right|}\mathbf{\le }\mathbf{\left|}{\mathit{a}}_{\mathbf{n}}\mathbf{\right|}$and$\underset{\mathbf{n}\mathbf{\to }\mathbf{\infty }}{\mathbf{l}\mathbf{i}\mathbf{m}}{\mathit{a}}_{\mathbf{n}}\mathbf{=}\mathbf{0}$, then an alternating series converges.

Finding convergence

Let us consider the hint, the sequence can be defined now:

$b_{\mathrm{n}}=a_{\mathrm{n}}+\left|{\mathrm{a}}_{\mathrm{n}}\right|$

so that. Also see that the following stands:

$\left|b_{\mathrm{n}}\right|\le 2\left|a_n\right|$

Now if we sum both sides and take into account thatlocalid="1658730771138" $\sum _{n=0}^{\infty }{a}_{\mathrm{n}}$series is absolutely convergent, we get:

localid="1658730630238" $\sum _{n=0}^{\infty }\left|{\mathrm{b}}_{\mathrm{n}}\right|=\sum _{n=0}^{\infty }{b}_{\mathrm{n}}\underline{<}2\rho$

Where $\rho$is defined as $\sum _{n=0}^{\infty }\left|a_{\mathrm{n}}\right|=\rho$. The first equality in the previous equation stands because $b_n\underline{>0}$.

Applying convergence

The series $\sum _{n=0}^{\infty }{b}_{\mathrm{n}}$is convergent. Now using the equation 1 can write:

localid="1658730886396" $\sum _{n=0}^{\infty }{a}_{\mathrm{n}}=\sum _{n=0}^{\infty }{b}_{\mathrm{n}}=\sum _{n=0}^{\infty }\left|{\mathrm{a}}_{\mathrm{n}}\right|$

Proved that $\sum _{n=0}^{\infty }{b}_{\mathrm{n}}$is convergent and $\sum _{n=0}^{\infty }\left|{\mathrm{a}}_{\mathrm{n}}\right|$is convergent by assumption so the $\sum _{n=0}^{\infty }{a}_{\mathrm{n}}$series must also converge.