Mathematica 8:

Series[(1/(a1))^(b)*(Cos[c*Log[1/(a1)]] + I*Sin[c*Log[1/(a1)]]), {c,

0, 6}] +

Series[(1/(a2))^(b)*(Cos[c*Log[1/(a2)]] + I*Sin[c*Log[1/(a2)]]), {c,

0, 6}] +

Series[(1/(a3))^(b)*(Cos[c*Log[1/(a3)]] + I*Sin[c*Log[1/(a3)]]), {c,

0, 6}] +

Series[(1/(a4))^(b)*(Cos[c*Log[1/(a4)]] + I*Sin[c*Log[1/(a4)]]), {c,

0, 6}] +

Series[(1/(a5))^(b)*(Cos[c*Log[1/(a5)]] + I*Sin[c*Log[1/(a5)]]), {c,

0, 6}] +

Series[(1/(a6))^(b)*(Cos[c*Log[1/(a6)]] + I*Sin[c*Log[1/(a6)]]), {c,

0, 6}] +

Series[(1/(a7))^(b)*(Cos[c*Log[1/(a7)]] + I*Sin[c*Log[1/(a7)]]), {c,

0, 6}]

The output is interesting.

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