E-GP03 · An exceptional zero first term
For a real parameter k, a sequence is defined by \(u_1=k+2\) and \(\displaystyle u_{n+1}=\frac{k-1}{2}u_n\). A zero first term and a zero ratio are allowed.
Assuming \(k\ne-2\), find the values of k for which the infinite series converges.
Now consider \(k=-2\) separately. Give the complete convergence set for all real k, and the sum for the exceptional case.
Find every value of k for which the sum to infinity equals 12.
Complete-answer checking, worked solutions and indicative written criteria.
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