Prove $\frac{1}{2}<S<1$ in Sequence Challenge

In summary, to prove $\frac{1}{2}<S<1$ in a sequence challenge, the sequence must be shown to converge to a value between $\frac{1}{2}$ and $1$ using techniques such as the squeeze theorem, the ratio test, and the root test. Proving this is important as it verifies the boundedness and convergence of the sequence. An example of a sequence that satisfies $\frac{1}{2}<S<1$ is the harmonic series. When proving this, common mistakes to avoid include assuming boundedness without justification, using incorrect methods, and neglecting to show convergence.
  • #1
anemone
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Given that $S=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4} + ...+\dfrac{1}{99}-\dfrac{1}{100}$.

Prove that $\dfrac{1}{2}<S<1$.
 
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  • #2
$$S = \left ( 1 - \frac{1}{2} \right ) + \left ( \frac{1}{3} - \frac{1}{4} \right ) + \cdots + \left ( \frac{1}{99} - \frac{1}{100} \right ) > \left ( 1 - \frac{1}{2} \right ) = \frac{1}{2}$$
$$S = 1 - \left ( \frac{1}{2} - \frac{1}{3} \right ) - \left ( \frac{1}{4} - \frac{1}{5} \right ) - \cdots - \frac{1}{100} < 1$$
$\blacksquare$
 
Last edited:
  • #3
Good job, Bacterius! And thanks for participating!
 

FAQ: Prove $\frac{1}{2}<S<1$ in Sequence Challenge

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