Show that ∑ n = k l 1 n 2 ≤ 1 k 2 + 1 k − 1 l holdsI'm aware that I could make use of convergences and solve this problem by estimating some Integrals.However, I want to use Calc 1 Methods, so basically I'm looking for some good estimates from above and below to proof the statement

Question

Show that       ∑          n      =      k              l            1          n      2        ≤      1          k      2        +      1    k    −      1    l   holdsI&#039;m aware that I could make use of convergences and solve this problem by estimating some Integrals.However, I want to use Calc 1 Methods, so basically I&#039;m looking for some good estimates from above and below to proof the statement

Clarence Mills · Accepted Answer

∑                      n            =            k                                l                                    1                      n            2                          =                  1                      k            2                          +                  ∑                      n            =            k            +            1                                l                                    1                      n            2                          ≤                  1                      k            2                          +                  ∑                      n            =            k            +            1                                l                                    1                      n            (            n            −            1            )                          =                  1                      k            2                          +                  ∑                      n            =            k            +            1                                l                                    (                      1                          n              −              1                                −                      1            n                    )                                    =                  1                      k            2                          +                  1          k                −                  1          l                .

reinzogoq · Accepted Answer

∑          n      =      k        l        1          n      2        =      1          k      2        +      ∑          n      =      k      +      1        l        1          n      2        ≤      1          k      2        +      ∑          n      =      k      +      1        l        (          1              n        −        1              −          1      n        )    =      1          k      2        +      1    k    −      1    l    .

Show that sum _(n=k )^l 1/(n^2)<= 1/(k^2) +1/k -1/l holds

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