anemone
Gold Member
MHB
POTW Director
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Hi MHB,
This problem vexes me until my mind hurts.
Problem:
Find the sum of the first 11 terms of the series $$\frac{19}{99}+\frac{199}{999}+\frac{1999}{9999}+ \cdots$$
Attempt:
I managed only to find the expression of the nth term of the given series and I got
$$\frac{19}{99}+\frac{199}{999}+\frac{1999}{9999}+ \cdots+11^{th} term=\sum_{k=1}^{11} \frac{2(10)^k-1}{10(10^k)-1}=\sum_{k=1}^{11} \left( \frac{1}{5}-\frac{4}{5(10(10^k)-1)} \right)$$
$$=\frac{11}{5}-\frac{4}{5} \left( \frac{1}{99}+\frac{1}{999}+ \cdots + 11^{th} term \right)$$
and I noticed this $$\sum_{k=1}^{11} \frac{4}{5(10(10^k)-1)} $$isn't a geometric series and thus by rewriting the problem in this manner is a dead end and it won't solve the problem.
So I tried to break the given series as follows:
$$\frac{19}{99}+\frac{199}{999}+\frac{1999}{9999}+ \cdots+11^{th} term=(\frac{1}{99}+\frac{9}{99})+(\frac{1}{999}+ \frac{2(99)}{999})+ \cdots+11th term$$
$$=\frac{1}{99}+\frac{1}{999}+ \cdots + 11^{th} term+2 \left( \frac{99}{999}+\frac{999}{9999}+ \cdots + 11^{th} term \right)$$and I know this is another cul-de-sac and I am getting so mad right now, I just don't see how to approach the problem.
Could anyone help me, please?
Thanks in advance.
This problem vexes me until my mind hurts.
Problem:
Find the sum of the first 11 terms of the series $$\frac{19}{99}+\frac{199}{999}+\frac{1999}{9999}+ \cdots$$
Attempt:
I managed only to find the expression of the nth term of the given series and I got
$$\frac{19}{99}+\frac{199}{999}+\frac{1999}{9999}+ \cdots+11^{th} term=\sum_{k=1}^{11} \frac{2(10)^k-1}{10(10^k)-1}=\sum_{k=1}^{11} \left( \frac{1}{5}-\frac{4}{5(10(10^k)-1)} \right)$$
$$=\frac{11}{5}-\frac{4}{5} \left( \frac{1}{99}+\frac{1}{999}+ \cdots + 11^{th} term \right)$$
and I noticed this $$\sum_{k=1}^{11} \frac{4}{5(10(10^k)-1)} $$isn't a geometric series and thus by rewriting the problem in this manner is a dead end and it won't solve the problem.
So I tried to break the given series as follows:
$$\frac{19}{99}+\frac{199}{999}+\frac{1999}{9999}+ \cdots+11^{th} term=(\frac{1}{99}+\frac{9}{99})+(\frac{1}{999}+ \frac{2(99)}{999})+ \cdots+11th term$$
$$=\frac{1}{99}+\frac{1}{999}+ \cdots + 11^{th} term+2 \left( \frac{99}{999}+\frac{999}{9999}+ \cdots + 11^{th} term \right)$$and I know this is another cul-de-sac and I am getting so mad right now, I just don't see how to approach the problem.
Could anyone help me, please?
Thanks in advance.