How can I find a formula for the sequence
$$10,\,110,\,1110,\, 11110,\dotsc$$
to make it ready for summation?
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2$a_n=\sum \limits_{k=1}^n\left(10^k\right)$ – Git Gud Nov 13 '13 at 19:00
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This is important to me sum 9/(10(10^n-1))=0.110918190.... – Nov 13 '13 at 19:16
3 Answers
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If we call your numbers $a_1,a_2,a_3,\dots$, then $$a_n=(10)\left(\frac{10^n-1}{9}\right).$$
André Nicolas
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Hint: at each step you are adding a power of $10$. That should make it easy to build a summation. Otherwise it is the sum of a finite geometric series.
Ross Millikan
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The sequence is defined by $$u_0=0,\quad u_n=10^n+u_{n-1}\ \forall n\ge1$$
so by telescoping and summing the geometric sequence we find $$u_n=\sum_{k=1}^n u_k-u_{k-1}=\sum_{k=1}^n10^k=10\frac{10^n-1}{10-1}$$