Seriesly
Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!
Age
16 to 18
Challenge level
Problem
Prove that
$$k \times k! = (k+1)! - k!$$ and sum the series
$$1 \times 1! + 2 \times 2! + 3 \times 3! +...+n \times n!$$
Did you know ... ?
A telecoping series is a series that can be written as the difference of two expressions in such a way that almost all the terms cancel with the following or preceding term leaving a few terms which can be combined to give the sum of the series. See the Wikipedia article Telescoping Series.
A telecoping series is a series that can be written as the difference of two expressions in such a way that almost all the terms cancel with the following or preceding term leaving a few terms which can be combined to give the sum of the series. See the Wikipedia article Telescoping Series.
Getting Started
When proving an identity it is usually best to start with the more complicated expression and simplify it so this time start with the expression on the right hand side $(k+1)! - k!$.
Student Solutions
This task previously appeared on the main
NRICH site, with solutions as follows:
Thank you for these solutions to Shahnawaz Abdullah; Daniel, Liceo Scientifico Copernico, Torino, Italy; Anderthan, Saratoga High School; Andrei, School 205, Bucharest, Romania; David, Queen Mary's Grammar School, Walsall; Paddy, Peter, Greshams School, Holt, Norfolk; Ngoc Tran, Nguyen Truong To High School (Vietnam); Chris, St. Bees School; Dorothy, Madras College; A Ji and Hyeyoun, St. Paul's Girls' School; and Yatir from Israel.
To prove that $k \times k! = (k+1)! - k!$. If we take $k!$ out as a factor from the right hand side of the equation, we are left with $k! \times ((k+1)-1)$ which simplifies to $k \times k!$, as required.
Now we sum the series $1 \times 1!+.....n \times n!$ As we have proved, $n \times n!$ is equal to $(n+1)! - n!$ and therefore $(n-1) \times (n-1)!$ is equal to $(n-1+1)! - (n-1)!$ which simplifies to $n! - (n-1)!$.
If we add the two results, we find that $n!$ cancels. If we sum the series from 1 to $n$, we find that all of the terms cancel except for $(n+1)!$ and $-(1!)$. Thus the sum of all numbers of the form $r \times r!$ from $1$ to $n$ is equal to $(n+1)! - 1$.
Thank you for these solutions to Shahnawaz Abdullah; Daniel, Liceo Scientifico Copernico, Torino, Italy; Anderthan, Saratoga High School; Andrei, School 205, Bucharest, Romania; David, Queen Mary's Grammar School, Walsall; Paddy, Peter, Greshams School, Holt, Norfolk; Ngoc Tran, Nguyen Truong To High School (Vietnam); Chris, St. Bees School; Dorothy, Madras College; A Ji and Hyeyoun, St. Paul's Girls' School; and Yatir from Israel.
To prove that $k \times k! = (k+1)! - k!$. If we take $k!$ out as a factor from the right hand side of the equation, we are left with $k! \times ((k+1)-1)$ which simplifies to $k \times k!$, as required.
Now we sum the series $1 \times 1!+.....n \times n!$ As we have proved, $n \times n!$ is equal to $(n+1)! - n!$ and therefore $(n-1) \times (n-1)!$ is equal to $(n-1+1)! - (n-1)!$ which simplifies to $n! - (n-1)!$.
If we add the two results, we find that $n!$ cancels. If we sum the series from 1 to $n$, we find that all of the terms cancel except for $(n+1)!$ and $-(1!)$. Thus the sum of all numbers of the form $r \times r!$ from $1$ to $n$ is equal to $(n+1)! - 1$.