Question
Download Solution PDF\({1\over2}+{3\over4}+{5\over8}+\ ...\infty\) का मान क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFअवधारणा:
समान्तर-ज्यामितीय श्रेणी (AGP):
- एक समान्तर-ज्यामितीय श्रेणी के पहले कुछ पद पहले पद a और सार्व अंतर d के साथ समान्तर श्रेणी से और पहले पद b और सार्व अनुपात r के साथ एक ज्यामितीय श्रेणी से बने हैं जिसे इसके द्वारा दिया जाता है:
ab, (a + d)br, (a + 2d)br2, ... [a + (n - 1)d]brn - 1, ...
-
एक AGP के अनंत का योग, जिसका |r| ≤ 1, निम्न द्वारा दिया गया है:
S∞ = \(\rm{ab\over1\ -\ r}\ +\ {dbr\over(1\ -\ r)^2}\)
गणना:
दी गई श्रृंखला \({1\over2}+{3\over4}+{5\over8}+\ ...\infty\) एक AGP है जिसमें
a = 1, d = 2 और b = \(1\over2\) , r = \(1\over2\)।
S∞ = \(\rm{1\left(1\over2\right)\over1-{1\over2}}+{(2)\left(1\over2\right)\left(1\over2\right)\over\left(1-{1\over2}\right)^2}\)
⇒ S∞ = 1 + \({1\over2}\over{1\over4}\)
⇒ S∞ = 1 + 2
⇒ S∞ = 3.
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