How do you use summation notation to expression the sum #15-3+3/5-...-3/625#?

Question

3654 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-03-07T10:11:24

Please see the explanation.

Let try to fill in the finite sum:

#15-3+3/5-3/25+3/125-3/625#

If your remove a common factor of 15, you get:

#1-1/5^1+1/5^2-1/5^3+1/5^4-1/5^5#

You can write 1 as #1/5^0#:

#1/5^0-1/5^1+1/5^2-1/5^3+1/5^4-1/5^5#

Because the minus sign appears on the odd powers, one can see that we are raising -5 to a negative power:

#1/5^0-1/5^1+1/5^2-1/5^3+1/5^4-1/5^5= sum_(n=0)^5-5^-n#

To obtain the original sum, multiply by 15:

#15-3+3/5-3/25+3/125-3/625=15sum_(n=0)^5-5^-n#