Prove the following series is true?

As n approaches infinity: 1/2+1/4+1/8+1/16+⋯+1/2^n =1 

8 Answers

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  • 4 weeks ago

    ½ = 1 - ½ {as ½ + ½ = 1}

    ½ + ¼ = 1 - ¼ {as ½ + ¼ + ¼ = 1}

    ½ + ¼ + ⅛ = 1 - ⅛ {etc.}

    ½ + ¼ + ⅛ + ¹∕₁₆ = 1 - ¹∕₁₆

    ½ + ¼ + ⅛ + ¹∕₁₆ + ¹∕₃₂ = 1 - ¹∕₃₂

    ½ + ¼ + ⅛ + ¹∕₁₆ + ¹∕₃₂ + … + ½ⁿ = 1 - ½ⁿ

    as n→∞, ½ⁿ→0, so we can discard the diminishing final term, yielding:[k = 1 to ∞]∑(½ᵏ) = 1

  • MyRank
    Lv 6
    4 weeks ago

    S = 1/2 + 1/4 + 1/8 + 1/10 + ………. + 1/2n = 1

    S = 1/2 + 1/(2)² + 1/(2)³ + ………… + 1/2.nΣ(S+1) = 1 + 1/2 + 1/(2)² + 1/(2)³ + 1/(2)⁴ + ……….S + 1 = 1 + 1/2 + 1/2² + 1/2³ + ……………..S + 1 = 1/2n.

  • Como
    Lv 7
    4 weeks ago

    :-

    S oo = a / ( 1 - r )

    S oo = (1/2) / ( 1 - 1/2 ) = 1

  • 4 weeks ago

    1/2 + 1/4 + 1/8 + 1/16 +⋯+1/2^n = 1 

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  • Let's say that this sums to S

    1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + .... = S

    S = 1/2 + (1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ....)

    S = 1/2 + 1/2 * (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ....))

    S = 1/2 + (1/2) * S

    S - (1/2) * S = 1/2

    S * (1 - 1/2) = 1/2

    S * (1/2) = 1/2

    S = 1

    S = 1/2 + 1/4 + 1/8 + 1/16 + ....

    S = 1

    Therefore

    1 = 1/2 + 1/4 + 1/8 + 1/16 + ....

  • oubaas
    Lv 7
    4 weeks ago

    sure : 1/4 +1/8 +1/16+1/32...+1/2^n = 1/2

    1/2+1/2 = 1.00

    • rotchm
      Lv 7
      4 weeks agoReport

      His solution is wrong since it assumes the result. U can always prove an assertion if you assume its true. You should have noticed this error. Look at my solution say; nowhere did I assume that the sum equals 1, yet I arrive at that conclusion. 

  • rotchm
    Lv 7
    4 weeks ago

    S = 1/2+1/4+1/8+1/16+⋯+1/2^n

    2S = 2(1/2+1/4+1/8+1/16+⋯+1/2^n)

    2S=1 + (1/2  + 1/4 + ... 1/2^(n-1)) 

    2S=1 + (1/2 + 1/4 + ... 1/2^(n-1) + 1/2ⁿ) - 1/2ⁿ

    2S= 1 + S - 1/2ⁿ

    Subtracting S each side gives

    S = 1 - 1/2ⁿ.

    If n is very large, what is S?

    Done!

  • JOHN
    Lv 7
    4 weeks ago

    S(n) = ½ + ¼ + 1/8 +.....+ 1/2ⁿ

    (1/2)S(n) = ¼ + 1/8 +.....+ 1/2ⁿ⁺¹

    S(n) - (1/2)S(n) = (1/2)S(n) = ½ - 1/2ⁿ⁺¹

    S(n) = 1 - 1/2ⁿ

    As n→∞, 1/2ⁿ⁺¹→0 & S(n) →1.

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