sum of reciprocal numbers of combinations











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Let $^nC_k:=dfrac{n!}{k!(n-k)!}$
Please prove that,for all natural number $k≥2$, $displaystylesum_{n=k+1}^{infty}frac{1}{^nC_k}=frac{1}{k-1}$



I tried to prove by induction, but I cannot. I guess it is proved by using Tayler series for some function, but I cannot find the function.










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    up vote
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    Let $^nC_k:=dfrac{n!}{k!(n-k)!}$
    Please prove that,for all natural number $k≥2$, $displaystylesum_{n=k+1}^{infty}frac{1}{^nC_k}=frac{1}{k-1}$



    I tried to prove by induction, but I cannot. I guess it is proved by using Tayler series for some function, but I cannot find the function.










    share|cite|improve this question


























      up vote
      2
      down vote

      favorite
      1









      up vote
      2
      down vote

      favorite
      1






      1





      Let $^nC_k:=dfrac{n!}{k!(n-k)!}$
      Please prove that,for all natural number $k≥2$, $displaystylesum_{n=k+1}^{infty}frac{1}{^nC_k}=frac{1}{k-1}$



      I tried to prove by induction, but I cannot. I guess it is proved by using Tayler series for some function, but I cannot find the function.










      share|cite|improve this question















      Let $^nC_k:=dfrac{n!}{k!(n-k)!}$
      Please prove that,for all natural number $k≥2$, $displaystylesum_{n=k+1}^{infty}frac{1}{^nC_k}=frac{1}{k-1}$



      I tried to prove by induction, but I cannot. I guess it is proved by using Tayler series for some function, but I cannot find the function.







      combinatorics taylor-expansion






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      share|cite|improve this question













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      share|cite|improve this question








      edited Nov 17 at 14:56









      Yadati Kiran

      1,239317




      1,239317










      asked Nov 17 at 14:44









      J.Rie

      183




      183






















          2 Answers
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          up vote
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          down vote



          accepted










          That is known as the German tank problem, and is one of the fundamental Binomial Identities.






          share|cite|improve this answer




























            up vote
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            This is very simple to prove. All you need to note is the following identity :
            $$
            frac 1{a(a+1)...(a+m)} = frac{1}{m}left( frac 1{a(a+1)...(a+m-1)} - frac 1{(a+1)...(a+m)}right)
            $$



            Now, consider the sum:
            $$
            sum_{n=k+1}^{k+m} frac 1{binom nk} = k! times sum_{i=1}^{i=m} frac{i!}{(k+i)!} \ = k! sum_{i=1}^{i=m}frac 1{(i+1)(i+2)...(i+k)} \ = k! sum_{i=1}^{i=m} frac{1}{k-1}left(frac{1}{(i+1)...(i+k-1)} - frac 1{(i+2)...(i+k)}right) \ = frac{k!}{k-1}sum_{i=1}^{i=m} left(frac{1}{(i+1)...(i+k-1)} - frac 1{(i+2)...(i+k)}right) \ = frac{k!}{k-1} left(frac 1{k!} - frac 1{(m+2)...(m+k)}right) \ = frac 1{k-1} - frac{k!}{(k-1)(m+2)...(m+k)}
            $$



            By letting $m$ go to infinity, the answer is clearly $frac 1{k-1}$, since the second term goes to zero.






            share|cite|improve this answer





















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              2 Answers
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              2 Answers
              2






              active

              oldest

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              active

              oldest

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              active

              oldest

              votes








              up vote
              1
              down vote



              accepted










              That is known as the German tank problem, and is one of the fundamental Binomial Identities.






              share|cite|improve this answer

























                up vote
                1
                down vote



                accepted










                That is known as the German tank problem, and is one of the fundamental Binomial Identities.






                share|cite|improve this answer























                  up vote
                  1
                  down vote



                  accepted







                  up vote
                  1
                  down vote



                  accepted






                  That is known as the German tank problem, and is one of the fundamental Binomial Identities.






                  share|cite|improve this answer












                  That is known as the German tank problem, and is one of the fundamental Binomial Identities.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 17 at 15:54









                  G Cab

                  17.1k31237




                  17.1k31237






















                      up vote
                      0
                      down vote













                      This is very simple to prove. All you need to note is the following identity :
                      $$
                      frac 1{a(a+1)...(a+m)} = frac{1}{m}left( frac 1{a(a+1)...(a+m-1)} - frac 1{(a+1)...(a+m)}right)
                      $$



                      Now, consider the sum:
                      $$
                      sum_{n=k+1}^{k+m} frac 1{binom nk} = k! times sum_{i=1}^{i=m} frac{i!}{(k+i)!} \ = k! sum_{i=1}^{i=m}frac 1{(i+1)(i+2)...(i+k)} \ = k! sum_{i=1}^{i=m} frac{1}{k-1}left(frac{1}{(i+1)...(i+k-1)} - frac 1{(i+2)...(i+k)}right) \ = frac{k!}{k-1}sum_{i=1}^{i=m} left(frac{1}{(i+1)...(i+k-1)} - frac 1{(i+2)...(i+k)}right) \ = frac{k!}{k-1} left(frac 1{k!} - frac 1{(m+2)...(m+k)}right) \ = frac 1{k-1} - frac{k!}{(k-1)(m+2)...(m+k)}
                      $$



                      By letting $m$ go to infinity, the answer is clearly $frac 1{k-1}$, since the second term goes to zero.






                      share|cite|improve this answer

























                        up vote
                        0
                        down vote













                        This is very simple to prove. All you need to note is the following identity :
                        $$
                        frac 1{a(a+1)...(a+m)} = frac{1}{m}left( frac 1{a(a+1)...(a+m-1)} - frac 1{(a+1)...(a+m)}right)
                        $$



                        Now, consider the sum:
                        $$
                        sum_{n=k+1}^{k+m} frac 1{binom nk} = k! times sum_{i=1}^{i=m} frac{i!}{(k+i)!} \ = k! sum_{i=1}^{i=m}frac 1{(i+1)(i+2)...(i+k)} \ = k! sum_{i=1}^{i=m} frac{1}{k-1}left(frac{1}{(i+1)...(i+k-1)} - frac 1{(i+2)...(i+k)}right) \ = frac{k!}{k-1}sum_{i=1}^{i=m} left(frac{1}{(i+1)...(i+k-1)} - frac 1{(i+2)...(i+k)}right) \ = frac{k!}{k-1} left(frac 1{k!} - frac 1{(m+2)...(m+k)}right) \ = frac 1{k-1} - frac{k!}{(k-1)(m+2)...(m+k)}
                        $$



                        By letting $m$ go to infinity, the answer is clearly $frac 1{k-1}$, since the second term goes to zero.






                        share|cite|improve this answer























                          up vote
                          0
                          down vote










                          up vote
                          0
                          down vote









                          This is very simple to prove. All you need to note is the following identity :
                          $$
                          frac 1{a(a+1)...(a+m)} = frac{1}{m}left( frac 1{a(a+1)...(a+m-1)} - frac 1{(a+1)...(a+m)}right)
                          $$



                          Now, consider the sum:
                          $$
                          sum_{n=k+1}^{k+m} frac 1{binom nk} = k! times sum_{i=1}^{i=m} frac{i!}{(k+i)!} \ = k! sum_{i=1}^{i=m}frac 1{(i+1)(i+2)...(i+k)} \ = k! sum_{i=1}^{i=m} frac{1}{k-1}left(frac{1}{(i+1)...(i+k-1)} - frac 1{(i+2)...(i+k)}right) \ = frac{k!}{k-1}sum_{i=1}^{i=m} left(frac{1}{(i+1)...(i+k-1)} - frac 1{(i+2)...(i+k)}right) \ = frac{k!}{k-1} left(frac 1{k!} - frac 1{(m+2)...(m+k)}right) \ = frac 1{k-1} - frac{k!}{(k-1)(m+2)...(m+k)}
                          $$



                          By letting $m$ go to infinity, the answer is clearly $frac 1{k-1}$, since the second term goes to zero.






                          share|cite|improve this answer












                          This is very simple to prove. All you need to note is the following identity :
                          $$
                          frac 1{a(a+1)...(a+m)} = frac{1}{m}left( frac 1{a(a+1)...(a+m-1)} - frac 1{(a+1)...(a+m)}right)
                          $$



                          Now, consider the sum:
                          $$
                          sum_{n=k+1}^{k+m} frac 1{binom nk} = k! times sum_{i=1}^{i=m} frac{i!}{(k+i)!} \ = k! sum_{i=1}^{i=m}frac 1{(i+1)(i+2)...(i+k)} \ = k! sum_{i=1}^{i=m} frac{1}{k-1}left(frac{1}{(i+1)...(i+k-1)} - frac 1{(i+2)...(i+k)}right) \ = frac{k!}{k-1}sum_{i=1}^{i=m} left(frac{1}{(i+1)...(i+k-1)} - frac 1{(i+2)...(i+k)}right) \ = frac{k!}{k-1} left(frac 1{k!} - frac 1{(m+2)...(m+k)}right) \ = frac 1{k-1} - frac{k!}{(k-1)(m+2)...(m+k)}
                          $$



                          By letting $m$ go to infinity, the answer is clearly $frac 1{k-1}$, since the second term goes to zero.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Nov 17 at 16:13









                          астон вілла олоф мэллбэрг

                          36.7k33376




                          36.7k33376






























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