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Limits of sequences

Limits

Japanese school year: Math III

What you learn

This topic examines the behavior of sequences as terms approach infinity, distinguishing between convergence to a limit and divergence. It provides essential tools for evaluating infinite geometric series and modeling discrete dynamical systems. Prior mastery of general sequence formulas, geometric progressions, and inequality manipulation is necessary for this unit.

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Key points

This formula shows that when the common ratio rr has an absolute value less than 1 (∣r∣<1|r| < 1), multiplying it repeatedly causes rnr^n to approach 0 as nn becomes infinite. It is used to analyze geometric sequences.

lim⁡n→∞rn=0(∣r∣<1)\lim_{n \to \infty} r^n = 0 \quad (|r| < 1)

This formula calculates the sum of an infinite geometric series with first term aa and common ratio rr. As long as ∣r∣<1|r| < 1, the infinitely many terms do not blow up, but add up to a1−r\frac{a}{1-r}.

∑n=1∞arn−1=a1−r(∣r∣<1)\sum_{n=1}^\infty a r^{n-1} = \frac{a}{1-r} \quad (|r| < 1)

This formula shows that as the denominator nn grows infinitely large, the fraction 1np\frac{1}{n^p} approaches 0 for any positive power p>0p > 0. It serves as a foundational rule for evaluating limits.

lim⁡n→∞1np=0(p>0)\lim_{n \to \infty} \frac{1}{n^p} = 0 \quad (p > 0)

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