Proving question, pattern and inequalitiesHi my math tutor gave me this problem to do over the week: prove 1/1^2+1/2^2+1/3^2...1/1000^2 &lt;2I've managed to almost complete the question, but I think I'm missing something or made a mistake.My attempt at the question: 1/1000^2 &lt; 1/999*1000 1/2^2+1/3^2+...1/1000^2&lt;1/1*2+1/2*3+...1/999*1000 1/1*2+1/2*3+...1/999*1000= (1-1/2)+(1/2-1/3)+(1/3-1/4)...+1/999-1/1000 (we can cancel out each fraction except for 1-1/1000)Therefore, 1/1*2+1/2*3+...1/999*1000=1-1/1000 1/2^2+1/3^2+...1/1000^2&lt;1-1000 1/1^2+1/2^2+1/3^2...1/1000^2&lt;2-1000As you can see, I'm stuck please help D;

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Proving question, pattern and inequalitiesHi my math tutor gave me this problem to do over the week: prove 1/1^2+1/2^2+1/3^2...1/1000^2 &amp;lt;2I&#039;ve managed to almost complete the question, but I think I&#039;m missing something or made a mistake.My attempt at the question: 1/1000^2 &amp;lt; 1/999*1000 1/2^2+1/3^2+...1/1000^2&amp;lt;1/1*2+1/2*3+...1/999*1000 1/1*2+1/2*3+...1/999*1000= (1-1/2)+(1/2-1/3)+(1/3-1/4)...+1/999-1/1000 (we can cancel out each fraction except for 1-1/1000)Therefore, 1/1*2+1/2*3+...1/999*1000=1-1/1000 1/2^2+1/3^2+...1/1000^2&amp;lt;1-1000 1/1^2+1/2^2+1/3^2...1/1000^2&amp;lt;2-1000As you can see, I&#039;m stuck please help D;

Reagan Madden · Accepted Answer

1          1      2        +      1          2      2        +      1          3      2        +  .  .  .  +      1          1000      2        &amp;lt;  &amp;lt;      1          1      2        +      1          1      ⋅      2        +      1          2      ⋅      3        +  .  .  .  +      1          999      ⋅      1000        =  =  1  +      1    1    −      1    2    +      1    2    −      1    3    +  .  .  .  +      1    999    −      1    1000    =  1  +  1  −      1    1000    &amp;lt;  2

Lydia Carey · Accepted Answer

By Riemann Sums:      1          1      2        +      1          2      2        +  ⋯  +      1          1000      2        =  1  +      (          1              2        2              +    ⋯    +          1              1000        2              )    ≤  1  +      ∫    1          1000                  d      x              x      2        =  1  +      999    1000    &amp;lt;  2

Proving question, pattern and inequalities Hi my math tutor gave me this problem to do over the wee

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