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Question Number 45599 by maxmathsup by imad last updated on 14/Oct/18
1)calculateAn=∫0n(−1)[x]2x+1−[x]dx2)findlimn→+∞An3)studytheserieΣAn
Commented by maxmathsup by imad last updated on 21/Oct/18
1)wehaveAn=∑k=0n−1∫kk+1(−1)k2x+1−kdx=∑k=0n−1(−1)k∫kk+1dx2x+1−k=∑k=0n−1(−1)k2[ln∣2x+1−k∣]kk+1=∑k=0n−1(−1k2{ln(k+3)−ln(k+1)}=∑k=0n−1(−1)k2ln(k+3k+1)⇒An=12{ln(3)−ln(2)+ln(53)−ln(64)+...(−1)n−1ln(n+2n)}
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