Extending the definition of the summation convention

In summary, the problem involves calculating the values of a_{i}b_{i} and a_{i}c_{i} for the given expressions, where i takes all integer values from 0 to ∞. It is also mentioned that the dot product is equivalent to the sum of a_{i}b_{i} from i=1 to i=3, but it is unclear how to incorporate the values of i. The problem is asking for the sums of xi/i! and (-1)ixi from i=1 to ∞.
  • #1

ppy

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Homework Statement



let a[itex]_{i}[/itex]=x[itex]^{i}[/itex] and b[itex]_{i}[/itex]=1[itex]\div[/itex] i ! and c[itex]_{i}[/itex]=(-1)[itex]^{i}[/itex] and suppose that i takes all interger values from 0 to ∞. calculate a[itex]_{i}[/itex]b[itex]_{i}[/itex] and calculate a[itex]_{i}[/itex]c[itex]_{i}[/itex]

Homework Equations


i know that in suffix notation a[itex]_{i}[/itex]b[itex]_{i}[/itex] is the same as the dot product as when you have to of the same subscripts you take the sum of a[itex]_{i}[/itex]b[itex]_{i}[/itex] from i=1 to i=3 but i am not really sure
of how to use the part where it says take interger values of i from 0 to ∞ an explanation would be great .
thanks.
 
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  • #2
hi ppy! :smile:
ppy said:
let a[itex]_{i}[/itex]=x[itex]^{i}[/itex] and b[itex]_{i}[/itex]=1[itex]\div[/itex] i ! and c[itex]_{i}[/itex]=(-1)[itex]^{i}[/itex] and suppose that i takes all interger values from 0 to ∞. calculate a[itex]_{i}[/itex]b[itex]_{i}[/itex] and calculate a[itex]_{i}[/itex]c[itex]_{i}[/itex]

it's asking you for ∑ xi/i! and ∑ (-1)ixi :wink:
 
  • #3
ppy said:
from i=1 to i=3

you actually run i from 1 to ##\infty## not 1 to 3
 

1. What is the summation convention and why is it important in science?

The summation convention is a mathematical notation used to simplify the writing of equations involving summations. It allows for compact and efficient representation of mathematical expressions involving repeated indices, making it a useful tool in many branches of science, including physics, mathematics, and engineering.

2. How can the definition of the summation convention be extended?

The definition of the summation convention can be extended by including more complex mathematical operations, such as integrals and derivatives, in addition to the traditional summation symbol. This allows for a more versatile and flexible notation that can be applied to a wider range of equations and problems.

3. What are the advantages of extending the definition of the summation convention?

Extending the definition of the summation convention allows for more concise and elegant mathematical expressions, reducing the amount of writing and making equations easier to read and understand. It also allows for a more general notation that can be applied to a wider variety of problems, making it a useful tool for scientists and mathematicians.

4. Are there any limitations to extending the definition of the summation convention?

One limitation of extending the definition of the summation convention is that it may become more complex and difficult to understand for those who are not familiar with the extended notation. It may also not be suitable for all types of equations and may not always provide a simpler representation compared to the traditional summation convention.

5. How can scientists learn more about the extended definition of the summation convention?

Scientists can learn more about the extended definition of the summation convention through textbooks, online resources, and discussions with colleagues. They can also practice using it in their own equations and seek feedback from others to improve their understanding and application of the extended notation.

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