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I am reading a book on financial theory and it has been a while since I took a notation heavy math course so could someone help me out here...

equation

Now with this equation how does the sum work? Do I do the factorials first? Is it a sum for each value from 0 to i? Is everything to the right of the sum sign included each iteration of the sum?

thanks!

Jackie
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1 Answers1

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In this particular case, yes, everything to the right of the summation sign is included in each iteration of the sum.

It is a sum for each value of i from 0 to N.

If you wanted to evaluate this at some value, you would do the summation first, and then begin to expand the resulting expression.

For example, consider the simpler expression

$$(1 + n^2)\sum\limits_{i = 1}^n i + 2.$$

evaluated at $n = 4$, say.

The first step to take would be to substitute in $4$ for $n$, giving

$$(1 + 4^2)\sum\limits_{i = 1}^4 i + 2.$$

Then, expand the summation, giving

$$\begin{align} (1 + 4^2)((1 + 2) + (2 + 2) + (3 + 2) + (4 + 2)) &= (1 + 16)(3 + 4 + 5 + 6)\\ &= (17)(18)\\ &= 306. \end{align}$$

Essentially, the summation is expanded first, then the interior of the summation is evaluated, and then that result gets combined with the outside.

You can think of a summation as implying parentheses, with the left paren just to the left of the $\sum$, and the right paren at the far end of the expression being summed over.

qaphla
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  • What about the N! that remains constant then right? – Jackie Aug 19 '13 at 19:08
  • Yes, the $N!$ will be a constant when this is evaluated at a given $N$. It will not be different in different terms of the summation. – qaphla Aug 19 '13 at 19:09
  • One last thing, any help on the h or the S at the end? Are these signifying functions or methods? Where would those be defined? – Jackie Aug 19 '13 at 19:16
  • The h and S appear to be functions that I would expect to have been defined somewhere earlier in your text. Not being an expert in finance, I can't really say what they each are. – qaphla Aug 19 '13 at 19:31