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Sum Fractional Ratios (Posted on 2023-08-07)

Determine the value of M when:

M = (3²+1)*(3²-1)^-1+(5²+1)*(5²-1)^-1+(7²+1)*(7²-1)^-1+......+ (999²
+1)*(999²-1)^-1

See The Solution Submitted by K Sengupta

Rating: 4.0000 (2 votes)

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Analytic Solution

| Comment 2 of 4 |

For each term (n^2+1)/(n^2-1), expand into [1 + 1/(n-1) - 1/(n+1)].

Then we have [1 + 1/2 - 1/4] + [1 + 1/4 - 1/6] + [1 + 1/6 - 1/8] + ... + [1 + 1/998 - 1/1000]

There are 499 terms, so them the +1 terms simplify to 499. What is left is a telescoping series, so that sum is 1/2-1/1000.

So the final sum is M=499+1/2-1/1000 = 499+499/1000 = 499.499.

Edited on August 8, 2023, 9:45 am

Posted by Brian Smith on 2023-08-07 13:50:47

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