(1/4 + 1/5)/(1 - 2/6 * 3/5)
Answers
Answer: An alternating infinite series having the sum log 2
Proof: The infinite series
is alternating since its terms are alternately positive and negative. Each term is numerically less than the preceding term and the terms decrease indefinitely. Putting it formally, nth term u(n) is monotonic decreasing, i.e. u(n) > u(n+1) for all values of n and u(n) = (-1)^(n-1)/n → 0 as n → ∞ . Hence the series is convergent. Since the series is convergent, it has a finite sum that can be obtained by putting x=1 in the Logarithmic Series (which is known to be convergent in the interval -1< x ≤1)
and the sum is log 2.
Alternatively, the sum can be determined using the result, as n → ∞, that
1+1/2+1/3+1/4+1/5++1/n - log n → γ
where γ is a constant (Euler’s constant). The above result can be rewritten as
where γ(n) → γ as n → ∞ . Let s(n) denote the algebraic sum of its first n terms; s the sum of the series. Then
= lim [log 2n + γ(2n) - log n -γ(n)] as n → ∞
= lim [log 2+ log n +γ(2n)- log n -γ(n)] as n→∞ which,on cancellation of log n term,
= lim [log 2+ + γ(2n) - γ(n)] as n→∞
→ log 2 + γ - γ [since γ(2n) - γ(n) → γ - γ = 0]
= log 2
There is yet another way we can arrive at the sum of the series (1). By the binomial theorem , if 0 ≤ x < 1, we have
1/(1+x) = (1+x)¯¹ = 1 - x + x² - x³ + ………,
Integrating between limits 0 to 1,
Since the latter series is convergent, we can integrate the L.H.S. and have
The sum log 2.
A fraction is equal parts of a whole thing, where numerator represents the number of equal parts and denominator represents equal parts of the whole thing.
To find the product we will multiply numerator with numerator and denominator with denominator.
i) a) 1/4 of 1/4 = 1/4 × 1/4 = 1/16
b) 1/4 of 3/5 = 1/4 × 3/5 = 3/20
c) 1/4 of 4/3 = 1/4 × 4/3 = 4/12 = 1/3 ( By reducing and simplifying the fraction)
ii) a) 1/7 of 2/9 = 1/7 × 2/9 = 2/63
b) 1/7 of 6/5 = 1/7 × 6/5 = 6/35
c) 1/7 of 3/10 = 1/7 × 3/10 = 3/70
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