Solving Summations of n.n! Terms

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In summary, the given series has a first term in an arithmetic progression with common difference of 1, and the nth term can be expressed as n.n! When expanded, the series becomes n.n(n-1)(n-2)...1. To compute the summation, we can consider n as (n+1)-1 and simplify to get (n+1)!.
  • #1
utkarshakash
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Homework Statement


1.1!+2.2!+3.3!...n terms

Homework Equations



The Attempt at a Solution


The first part of every term is in AP whose cd is 1. Also the nth term of this series is given as n.n! If I expand it it becomes n.n(n-1)(n-2)...1

Now
[itex]S_n=\sum t_n \\
= \sum n.n(n-1)(n-2)...1 [/itex]

But now how do I compute this summation?
 
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Consider n = (n+1)-1
 
  • #3
Mentallic said:
Consider n = (n+1)-1

I didn't get you. What you said was to replace all the n's with (n+1)-1 but I didn't find it to be helpful. Can you please give me some more hints?
 
  • #4
utkarshakash said:
I didn't get you. What you said was to replace all the n's with (n+1)-1 but I didn't find it to be helpful. Can you please give me some more hints?

Not all n's!

n.n! = [(n+1)-1]n!
 

FAQ: Solving Summations of n.n! Terms

What is a summation of n.n! terms?

A summation of n.n! terms is a mathematical expression that involves adding a series of terms where each term is the product of a number, n, and its factorial, n!. For example, a summation of 5.5! terms would be expressed as 5.5! + 4.4! + 3.3! + 2.2! + 1.1!.

What is the purpose of solving summations of n.n! terms?

The purpose of solving summations of n.n! terms is to find the total value of the expression. These types of summations can arise in various mathematical and scientific problems, and solving them can help in understanding and solving more complex equations.

What are some common techniques for solving summations of n.n! terms?

Some common techniques for solving summations of n.n! terms include using mathematical properties such as the distributive property, simplifying the expressions by grouping terms, and using formulas for calculating factorials. Another helpful technique is to look for patterns in the terms and use that to simplify the expression.

What are some challenges that may arise when solving summations of n.n! terms?

One challenge that may arise when solving summations of n.n! terms is dealing with large numbers and complex expressions. Another challenge is identifying the correct formula or technique to use for a particular summation. Additionally, it is important to be careful with calculations and avoid mistakes, as they can significantly impact the final result.

How can solving summations of n.n! terms be applied in real-life situations?

Solving summations of n.n! terms can be applied in various real-life situations, such as in statistics, probability, and physics. For example, it can be used to calculate the number of possible outcomes in a series of events or to find the total energy in a system with multiple components. It can also be applied in computer science and programming to optimize algorithms and solve problems involving large numbers and factorial calculations.

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