Chứng minh rằng 1/4+1/16+1/36+1/64+…+1/196<1/2 04/10/2021 Bởi Melanie Chứng minh rằng 1/4+1/16+1/36+1/64+…+1/196<1/2
TL: $\frac{1}{4}$ $+ \frac{1}{16}$ $+ \frac{1}{64}$ $+ \frac{1}{100}$ $+ \frac{1}{144}$ $+\frac{1}{196}$ $< \frac{1}{2}$ $= \frac{1}{4}$ $+ \frac{1}{16}$ $+ \frac{1}{64}$ $+ \frac{1}{100}$ $+ \frac{1}{144}$ $+\frac{1}{196}$ $< \frac{1}{2^2 – 1 }$ $+ \frac{1}{4^2 – 1}$ $+ \frac{1}{6^2 – 1}$ $+ … +\frac{1}{14^2 – 1}$ $= \frac{1}{1.3}$ $\frac{1}{3.5}$ $\frac{1}{5.7}$ $+ … +\frac{1}{13.15}$ $=\frac{1}{2}$ $( 1 – \frac{1}{3}$ $+\frac{1}{3}$ $- $ …. $ – \frac{1}{13}$ $-\frac{1}{13}$ $+\frac{1}{15} )$ $=\frac{1}{2}( 1 – $ $\frac{1}{15})$ $<\frac{1}{2}$ $Vậy $ $\frac{1}{4}$ $+ \frac{1}{16}$ $+ \frac{1}{64}$ $+ \frac{1}{100}$ $+ \frac{1}{144}$ $+\frac{1}{196}$ $< \frac{1}{2}$ Bình luận
TL: 1414 +116+116 +164+164 +1100+1100 +1144+1144 +1196+1196 <12<12 =14=14 +116+116 +164+164 +1100+1100 +1144+1144 +1196+1196 <122−1<122−1 +142−1+142−1 +162−1+162−1 +...+1142−1+…+1142−1 =11.3=11.3 13.513.5 15.715.7 +...+113.15+…+113.15 =12=12 (1−13(1−13 +13+13 −− …. −113−113 −113−113 +115)+115) =12(1−=12(1− 115)115) <12<12 VậyVậy 1414 +116+116 +164+164 +1100+1100 +1144+1144 +1196+1196 <12<12 Bình luận
TL:
$\frac{1}{4}$ $+ \frac{1}{16}$ $+ \frac{1}{64}$ $+ \frac{1}{100}$ $+ \frac{1}{144}$ $+\frac{1}{196}$ $< \frac{1}{2}$
$= \frac{1}{4}$ $+ \frac{1}{16}$ $+ \frac{1}{64}$ $+ \frac{1}{100}$ $+ \frac{1}{144}$ $+\frac{1}{196}$ $< \frac{1}{2^2 – 1 }$ $+ \frac{1}{4^2 – 1}$ $+ \frac{1}{6^2 – 1}$ $+ … +\frac{1}{14^2 – 1}$
$= \frac{1}{1.3}$ $\frac{1}{3.5}$ $\frac{1}{5.7}$ $+ … +\frac{1}{13.15}$
$=\frac{1}{2}$ $( 1 – \frac{1}{3}$ $+\frac{1}{3}$ $- $ …. $ – \frac{1}{13}$ $-\frac{1}{13}$ $+\frac{1}{15} )$
$=\frac{1}{2}( 1 – $ $\frac{1}{15})$ $<\frac{1}{2}$
$Vậy $ $\frac{1}{4}$ $+ \frac{1}{16}$ $+ \frac{1}{64}$ $+ \frac{1}{100}$ $+ \frac{1}{144}$ $+\frac{1}{196}$ $< \frac{1}{2}$
TL:
1414 +116+116 +164+164 +1100+1100 +1144+1144 +1196+1196 <12<12
=14=14 +116+116 +164+164 +1100+1100 +1144+1144 +1196+1196 <122−1<122−1 +142−1+142−1 +162−1+162−1 +...+1142−1+…+1142−1
=11.3=11.3 13.513.5 15.715.7 +...+113.15+…+113.15
=12=12 (1−13(1−13 +13+13 −− …. −113−113 −113−113 +115)+115)
=12(1−=12(1− 115)115) <12<12
VậyVậy 1414 +116+116 +164+164 +1100+1100 +1144+1144 +1196+1196 <12<12