Partial Sums

  1. 1. The problem statement, all variables and given/known data
    Let an (read 'a sub n') be the nth digit after the decimal point in 2pi+2e. Evaluate

    SUM (n=1 to inf) an(.1)^n

    (here, again, an is meant to be 'a sub n')

    2. Relevant equations

    As far as I can see, this is a partial sum of a geometric series. To find the nth partial sum (or, in other words the infinite sum) use a/(1-r) where a is the first term of the series (scalar multiple) and r is the ratio of the exponent of the general form for geo series: ar^n

    3. The attempt at a solution
    My attempt gave back to sn=1/9

    I used: .1/1-.1
    I guess I'm thinking about this incorrectly
  2. jcsd
  3. Rather than applying formulas, you need to stop and think for a second.

    What does it mean that [tex]a_n[/tex] is the [tex]n[/tex]th digit after the decimal point in the decimal expansion of [tex]2\pi + 2e[/tex] ? What is a decimal expansion?

    What is [tex]1 \cdot (0.1)^1 + 4 \cdot (0.1)^2 + 1 \cdot (0.1)^3 + 5 \cdot (0.1)^4 + 9 \cdot (0.1)^5[/tex]?
  4. A decimal expansion is the division of a rational expression p/q.
    To be an nth digit after the decimal point means that the rational expression, when divided will not have a finite number of decimal places.

  5. You need to think about the relationship between the digits of the decimal expansion and the number which is represented by the decimal expansion. This relationship can be expressed as an equation.
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