MathLabs

Problem 5

Let a1,…,a8a_1,\ldots,a_8 be real numbers, not all zero, and let cn=a1n+⋯+a8nc_n=a_1^n+\cdots+a_8^n for n=1,2,…n=1,2,\ldots. Suppose infinitely many terms cnc_n are zero. Determine all values of nn for which cn=0c_n=0.
Step 2 of 7: Suppress the smaller terms
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∑∣ai∣<R∣ai∣n<Rn\sum_{|a_i|<R}|a_i|^n<R^n
Detailed analysis

For all sufficiently large nn, the sum of the at most seven smaller absolute powers is less than RnR^n. Infinitely many zero indices are arbitrarily large, so choose such an index nn with cn=0c_n=0.