${G}$ = $\frac{1}{2^{2}}$+$\frac{1}{4^{2}}$+$\frac{1}{6^{2}}$+...+$\frac{1}{100^{2}}$. $\text{CMR}$ ${G}$ < $\frac{1}{2}$
Ta có: ${G}$ = $\frac{1}{2^{2}}$+$\frac{1}{4^{2}}$+$\frac{1}{6^{2}}$+...+$\frac{1}{100^{2}}$
⇒${G}$ = $\frac{1}{2^{2}}$(${1}$+$\frac{1}{2^{2}}$+$\frac{1}{3^{2}}$+...+$\frac{1}{50^{2}}$)
Lại có:
$\frac{1}{2^{2}}$ < $\frac{1}{1.2}$
$\frac{1}{3^{2}}$ < $\frac{1}{2.3}$
......
$\frac{1}{50^{2}}$ < $\frac{1}{49.50}$
⇒${G}$ = $\frac{1}{4}$(${1}$+$\frac{1}{2^{2}}$+$\frac{1}{3^{2}}$+...+$\frac{1}{50^{2}}$) < $\frac{1}{4}$(${1}$+$\frac{1}{1.2}$+$\frac{1}{2.3}$+...+$\frac{1}{49.50}$)
⇒${G}$ < $\frac{1}{4}$.( $\frac{1}{1}$-$\frac{1}{2}$+$\frac{1}{2}$-$\frac{1}{3}$+...+$\frac{1}{49}$-$\frac{1}{50}$)
⇒${G}$ < $\frac{1}{4}$.( $\frac{1}{1}$-$\frac{1}{50}$)
⇒${G}$ < $\frac{99}{100}$ < $\frac{100}{200}$ = $\frac{1}{2}$
⇒${G}$ < $\frac{1}{2}$
$\text{CHÚC BẠN HỌC TỐT!}$
