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Is it possible to find a suitable lower bound on $\left|\left(tM+\sum_{k=2}^\infty\frac{(tM)^k}{k!}\right)\cdot b\right|$ for $M$ as $n \times n$ matrix, $b$ as $1 \times n$ vector and for all $t$ in the form of $|tAb|-c|b|$ with appropriate $c$?

Or are there other practical approximations on the upper equation?

Thanks!

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    There is no particular reason why$I$distinguish k=1. I thought it would be helpful, because I wan to calculate the interception point of the upper function f(t)=|...| with an onther curve. I hoped to seperate the t by distinguishing k=1. Note, the solution should be less than or equal the upper function.2012-08-24

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