Let $a$ be an integer number greater than one, $\{k_i\}_{1≤i≤r}$ and $\{m_j\}_{1≤j≤s}$ two strictly increasing sequences of integer numbers. Suppose that $$\sum_{i=1}^r\frac{1}{a^{k_i}}=\sum_{j=1}^s\frac{1}{a^{m_j}}$$ Prove that $r = s$ and for each $1 \leq i \leq r$, exists $1 \leq j \leq r$ such that $k_i = m_j$. Any suggest(hint)? I don't want the answer. Thanks!
Questions about finite sums, calculate that you have the same amount of elements and also equals.
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real-analysis
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1Hint: try some examples for $a = 10$. – 2017-02-15
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0"Let $a$ be an integer greater than $1$" is not an example of a *question*. (Or of a topic.) – 2017-02-16
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0@MeesdeVries Someone told me, dial why is 0.011 ≠ 0.101 in base 10? I am a beginner in matters of real analysis. – 2017-02-17
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0Uh. Well. What do you mean by "why"? Those numbers denote, respectively, $10^{-2} + 10^{-3}$ and $10^{-1} + 10^{-3}$. If they were equal then so would $10^{-2}$ and $10^{-1}$ be, and thus so would $10$ and $1$ be. Is that enough? – 2017-02-17
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Assume WLOG that $k_r\ge m_s$ and multiply the equality by $a^{k_r}$. Then use congruences modulo $a^j_k$ for $j=r,r-1,\ldots,1$. Perhaps induction helps.
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0Can you explain better? @ajotatxe please – 2017-02-21