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The implication
is not generally valid for all
. It holds true only when
.
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Bonus Knowledge
Did you know that this inequality relates to the concept of geometric series? When you encounter expressions like
, it reminds us of the formula used to sum a geometric series. This comes in handy in various applications, like calculating the total of repeated investments or savings over time!
In calculus, understanding such inequalities is crucial for proving convergence and divergence of series. One common mistake when rearranging inequalities is neglecting the conditions of each term. Always double-check if
is greater than one for the terms to remain valid and avoid potential pitfalls in algebraic manipulation. Remember, mathematical inequalities can feel like a tightrope walk—stay balanced!

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