Synonymous series?
In school, Timmy was told to calculate the value of the series
a + a 2 + a 3 + a 4 + ⋯ ,
for some known a . But Timmy misheard, he instead calculated the series
a − 1 + a − 2 + a − 3 + a − 4 + ⋯ .
Surprisingly, Timmy gets the correct answer!
Assuming that he did his work correctly, what was his answer?
Submit you answer as -1000 if you think that this is an impossible scenario.
The answer is -1000.
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1 solution
No, that's not the right approach.
Just because a 1 + a 2 + a 3 + ⋯ = b 1 + b 2 + b 3 + ⋯ , that doesn't mean that a i = b i for i = 1 , 2 , 3 , 4 , … must be true.
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I'm not quite sure what you mean. I didn't imply that all terms are necessarily equal because the infinite sums are. I was referring to the closed forms of the two series: a/(1-a) and (1/a)/(1-(1/a)) (=1/(a-1)) and that equating to two ultimately results in a^2=1 which means that both series' diverge.
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Whoops, I glanced at your solution and misread the entire thing. Sorry.
Yup, you're right after all.
Additional challenge: If both series diverge to infinity, wouldn't that make both these series' equal?
By expressing the two series as rational functions and equating the two, it can be found that "a", which is both the initial term and common ratio for the first sequence (and correspondingly the reciprocal of both for the other) would be plus or minus 1, and an infinite geometric series with a common ratio of magnitude equal to or greater than 1 must be divergent.