Use sigma notation for expression

[synergix]

1. The problem statement, all variables and given/nknown data
Find a formula for the expression. Use sigma notation.
(x + 1/y)ⁿ

The Attempt at a Solution

Not sure how to do this we have only dealt with sigma notation involving only n as a variable

 

[dx]

Use the binomial theorem to expand the expression.

Not sure how to do this we have only dealt with sigma notation involving only n as a variable

Example:

$$\sum_{n=1}^{5} x^n = x + x^2 + x^3 + x^4 + x^5$$​

Notice that although x is a variable, it does not vary from term to term. The only thing that varies from term to term is the exponent. Your problem is similar.

 

[Architeuthis Dux]

The expression (x + 1/y)n can be rewritten as the sum of terms, where each term is (x + 1/y) raised to the nth power. This can be represented in sigma notation as:

∑(x + 1/y)^n

Where the summation is taken from n = 1 to n = ∞, or to any other specified limit.

Alternatively, if you are looking for a specific formula for this expression, it can be written as:

(x + 1/y)^n = ∑(x + 1/y)^n * n!/((n-k)!k!)

Where k is the index of summation, varying from 0 to n. This formula can also be simplified to:

(x + 1/y)^n = ∑(n+k)!/(n!k!) * x^(n-k) * (1/y)^k

I hope this helps! Let me know if you have any further questions.

 

[Architeuthis Dux]

.

To represent this expression using sigma notation, we can use the summation symbol ∑ and the index variable i. The formula would be:

∑ (x + 1/y)^n

Where i represents the term number in the expression. For example, the first term would be (x + 1/y)^1, the second term would be (x + 1/y)^2, and so on. The upper limit of the summation would depend on the context of the problem.

Alternatively, if the expression is given as a sum of terms, we can use the following formula:

∑ (x + 1/y)^n = ∑ (x^n) + ∑ (1/y^n)

Where the first summation represents the sum of all the x terms and the second summation represents the sum of all the 1/y terms.

Overall, the use of sigma notation allows us to represent and manipulate expressions in a more concise and organized manner.