If f ( x ) = e − x + 2 e − 2 x + 3 e − 3 x + ⋯ then find ∫ ln ⁡ 2 ln ⁡ 3 f ( x ) d x

Question

If   f  (  x  )  =      e          −      x        +  2      e          −      2      x        +  3      e          −      3      x        +  ⋯ then find       ∫          ln      ⁡      2              ln      ⁡      3        f  (  x  )  d  x

Aleah Rowe · Accepted Answer

Start with (bear with me for a moment):                              ∑                      n            ≥            0                                    z          n                                    =                  1                      1            −            z                                  |        z        |        &amp;lt;        1                                      ∑                      n            ≥            0                          n                  z                      n            −            1                                              =                  d                      d            z                                    1                      1            −            z                          =                  1                      (            1            −            z                          )              2                                                                    ∑                      n            ≥            1                          n                  z          n                                    =                  z                      (            1            −            z                          )              2                                          In your case,   z  =      e          −      x      , so for the limits of your integral       e          −      ln      ⁡      2        =      1    2   and       e          −      ln      ⁡      3        =      1    3  , which are comfortably inside the convergence range. So you can write:  f  (  x  )  =            e              −        x                    (      1      −              e                  −          x                            )        2            So your integral is:                              ∫                      ln            ⁡            2                                ln            ⁡            3                          f        (        x        )                d        x                            =                  ∫                      ln            ⁡            2                                ln            ⁡            3                                                              e                              −                x                                                  d            x                                (            1            −                          e                              −                x                                                    )              2                                                                        =        −                  ∫          2          3                                      d            u                                (            1            −            u                          )              2                                                                        =                                                        1                              1                −                u                                      |                                u            =            2                    3                                                  =                  1                      −            2                          −                  1                      −            1                                                            =                  1          2

Scott Valenzuela · Accepted Answer

Integration term by term is valid because the convergence is uniform by the Weierstrass M-test, hence      ∫          ln      ⁡      2              ln      ⁡      3        f  (  x  )  d  x  =      ∫          ln      ⁡      2              ln      ⁡      3            ∑          n      =      1        ∞    n      e          −      n      x        d  x  =      ∑          n      =      1        ∞    n      ∫          ln      ⁡      2              ln      ⁡      3            e          −      n      x        d  x  =      ∑          n      =      1        ∞        2          −      n        −      3          −      n        =      1          2      −      1        −      1          3      −      1        =      1    2    .

If f(x)=e^(-x)+2e^(-2x)+3e^(-3x)+... then find int_(ln 2)^(ln 3) f(x)dx

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Answer & Explanation

New Questions in Calculus 1