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The following screenshot is taken from some lecture notes on actuarial mathematics:


notes


As you can see, the notation $h \ll 1$ is used here (in the first line of the proof).

Does anyone know what this is supposed to denote? I first suspected that it was a typo and was instead supposed to be $h < 1$ or $h \leq 1$, but the same notation is used on several occasions (leading me to believe that its use must be intentional).

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It means “sufficiently smaller than”. So, $h\ll1$ means that $h$ is sufficiently smaller than $1$ (but still greater than $0$).

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It means $h$ is much smaller than $1$. It can be an informal way of working with powers, where you say you can neglect terms with a higher power of $h$ because they are even smaller. For example, we know $(1+h)^3= 1 + 3 h + 3 h ^ 2 + h ^ 3 $. If $h\ll1$ the terms with $h^2$ and $h^3$ are even smaller, so we may be able to neglect them.

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