So sánh

0

01263812493

CMR
[TEX] \frac{1}{5 ^ 2} + \frac{1}{6 ^ 2}+ \frac{1}{7 ^ 2}+...+ \frac{1}{100 ^ 2 } < \frac{1}{4}[/TEX]

[TEX]\frac{1}{5^2}<\frac{1}{4.5}[/TEX]

[TEX]\frac{1}{6^2}<\frac{1}{5.6}[/TEX]

[TEX]\frac{1}{7^2}<\frac{1}{6.7}[/TEX]

[TEX]................................................[/TEX]

[TEX]\frac{1}{100^2}<\frac{1}{99.100}[/TEX]

[TEX]\Rightarrow[/TEX]

[TEX]\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}+...+ \frac{1}{100^2}< \frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+.......+ \frac{1}{99.100}[/TEX]

[TEX]=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+.....+ \frac{1}{99}-\frac{1}{100}=\frac{1}{4}-\frac{1}{100}<\frac{1}{4}[/TEX]
 
W

w41_t13z

CMR
[TEX] \frac{1}{6}< \frac{1}{5 ^ 2} + \frac{1}{6 ^ 2}+ \frac{1}{7 ^ 2}+...+ \frac{1}{100 ^ 2 } < \frac{1}{4}[/TEX]

[TEX] \frac{1}{2!}+ \frac{1}{3!}+...+ \frac{1}{100!}< 1[/TEX]

Bài này làm tương tự như bài trên sẽ được:

[TEX] \frac{1}{2!}+ \frac{1}{3!}+...+ \frac{1}{100!}<\frac{1}{1.2}+ \frac{1}{2.3}+...+ \frac{1}{99.100} [/TEX] = [TEX]1- \frac{1}{100}[/TEX] < 1

 
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