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I'm taking a calculus class, but I skipped school the past week due to health problems. I spoke to my teacher and classmates and they told me that they had seen power series topic. So I got a copy of the notes, but I can't still figure out the relation of the series with the sequence

Ex.

  • 0,0,...,0,1,0, ... = X^n
  • 1,1,1,1,1,1,1,1,... = 1/(1-X)
  • 1,2,3,4,5,6,7, ... = 1/(1-X)^2

I know that each of the terms is multiplied by the X^n term, but I still don't get the idea

Thanks in advance for your help!

Best Regards!

Roosembert Palacios
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1 Answers1

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It looks like this is a notation defined by $$(a_0,a_1,a_2,\ldots) \leftrightarrow a_0 +a_1x+a_2x^2+\cdots$$ or, more briefly, $$(a_n)_{n\in I} \leftrightarrow \sum_{n\in I}a_nx^n$$ where $I$ is the set of nonnegative integers.

So your first example is a single term, the second example is a geometric series, and the third term is the derivative of a geometric series (you should verify the last statement by taking the derivative of the geometric series term-by-term and looking at the resulting coefficients).

The symbol $x$ is an indeterminate and is just used to distinguish the "slots" or coordinates of the expression. You use it all the time and don't realize it. For example, you think that two vectors are equal if and only if each if the corresponding coordinates are equal. You think that two polynomials are equal if and only if coefficients of corresponding powers of the indeterminate are equal. Same thing, different notation.

MPW
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