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
Gold Member
MHB
POTW Director
3,883
115
Given that $S=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4} + ...+\dfrac{1}{99}-\dfrac{1}{100}$.

Prove that $\dfrac{1}{2}<S<1$.
 
Mathematics news on Phys.org
  • #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

How do you prove that $\frac{1}{2}

In order to prove $\frac{1}{2}

What is the importance of proving $\frac{1}{2}

Proving $\frac{1}{2}

What are some common techniques used to prove $\frac{1}{2}

Some common techniques used to prove $\frac{1}{2}

Can you provide an example of a sequence that satisfies $\frac{1}{2}

One example of a sequence that satisfies $\frac{1}{2}

What are some common mistakes to avoid when proving $\frac{1}{2}

Some common mistakes to avoid when proving $\frac{1}{2}

Similar threads

MHB Sequence Challenge: Prove $a_{50}+b_{50}>20$
Replies
1
Views
1K
MHB Roots of Polynomial: Find $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$
Replies
1
Views
978
MHB Proving Series Inequality: $\sqrt[3]{\frac{2}{1}}$ to $\frac{1}{8961}$
Replies
1
Views
1K
MHB Proving $(a)^\dfrac{1}{3}+(b)^\dfrac{1}{3}+(c)^\dfrac{1}{3}=0$
Replies
1
Views
1K
MHB Prove $\frac{1}{3}\le \frac{u}{v} \le \frac{1}{2}$ with Triangle Sides
Replies
4
Views
1K
MHB New Year Challenge: Find Real Number Triples
Replies
1
Views
863
MHB Can You Prove this Trigonometric Inequality?
Replies
1
Views
930
MHB Can you prove this inequality challenge?
Replies
1
Views
874
MHB Proof: $\cos\dfrac{\pi}{7}=\dfrac{1}{6}+\dfrac{\sqrt{7}}{6}$
Replies
1
Views
862
MHB Algebra Challenge: Proving $p,\,q,\,r,\,s,\,t$ Satisfy Equation
Replies
1
Views
905

Hot Threads

Recent Insights

Back
Top