Does this series converge?

1/1!+1/3!+1/5!+.... infinity.

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  • Favorite Answer

    Yes.

    1/1! + 1/2! + 1/3! + 1/4! + .... converges to e

    1/1! + 1/3! + 1/5! + 1/7! + .... is less than the previous series, so it must converge as well

    1/1! + 1/2! - 1/2! + 1/3! + 1/4! - 1/4! + 1/5! + 1/6! - 1/6! + 1/7! + .... =>

    1/1! + 1/2! + 1/3! + 1/4! + 1/5! + .... - (1/2! + 1/4! + 1/6! + ....) =>

    e - (1/(2 * 1)! + 1/(2 * 2)! + 1/(2 * 3)! + ....)

    e - (cosh(1) - 1) =>

    e - (e^(1) + e^(-1)) / 2  -  1 =>

    e - (1/2) * e - 1/(2e) - 1 =>

    (1/2) * e - (1/2) * e^(-1)  -  1 =>

    (1/2) * (e^2  -  1) / e  -  1 =>

    (e^2 - 1 - 2e) / (2e) =>

    (e^2 - 2e + 1 - 2) / (2e) =>

    ((e - 1)^2 - 2) / (2e)

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  • 1 month ago

    Yes, it does converge, if you are adding an infinite number of terms rather than adding infinity.

    1/1! = 1/2⁰, 1/3! = 1/(1×2×3) < 1/2¹, 1/5! = 1/(1×2×3×4×5) < 1/2², ..., 1/(2n-1)! < 1/2⁽ⁿ⁻¹⁾, ...

    so 1/1! + 1/3! + 1/5! + ... < 1/2⁰ + 1/2¹ + 1/2² + ... which converges to 2

    so 1/1! + 1/3! + 1/5! + ...  converges (to something between 1 and 2)

    • JOHN
      Lv 7
      1 month agoReport

      It converges to sinh1 =1.175201194.

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