Pertanyaan

Nilai dari 2015 ! 1 + k = 1 ∑ 2014 ( k + 1 ) ! k adalah ...

Nilai dari $\frac{1}{2015 !} + k = 1 \sum 2014 \frac{k}{( k + 1 ) !}$ adalah ...

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Habis dalam

Klaim

I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Pembahasan

Perhatikan Bahwa maka Sehingga Jadi, nilai dari adalah 1

$fraction numerator 1 over denominator 2015 factorial end fraction plus sum from k equals 1 to 2014 of fraction numerator k over denominator open parentheses k plus 1 close parentheses factorial end fraction$

Perhatikan Bahwa

$table attributes columnalign right center left columnspacing 0px end attributes row cell sum from k equals 1 to 2014 of fraction numerator k over denominator open parentheses k plus 1 close parentheses factorial end fraction end cell equals cell sum from k equals 1 to 2014 of open parentheses fraction numerator 1 over denominator k factorial end fraction minus fraction numerator 1 over denominator open parentheses k plus 1 close parentheses factorial end fraction close parentheses end cell row blank equals cell open parentheses fraction numerator 1 over denominator 1 factorial end fraction minus fraction numerator 1 over denominator 2 factorial end fraction close parentheses plus open parentheses fraction numerator 1 over denominator 2 factorial end fraction minus fraction numerator 1 over denominator 3 factorial end fraction close parentheses plus horizontal ellipsis plus open parentheses fraction numerator 1 over denominator 2014 factorial end fraction minus fraction numerator 1 over denominator 2015 factorial end fraction close parentheses end cell row blank equals cell fraction numerator 1 over denominator 1 factorial end fraction minus fraction numerator 1 over denominator 2015 factorial end fraction end cell end table$

maka

$sum from k equals 1 to 2014 of fraction numerator k over denominator open parentheses k plus 1 close parentheses factorial end fraction equals 1 minus fraction numerator 1 over denominator 2015 factorial end fraction$

Sehingga

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator 2015 factorial end fraction plus sum from k equals 1 to 2014 of fraction numerator k over denominator open parentheses k plus 1 close parentheses factorial end fraction end cell equals cell fraction numerator 1 over denominator 2015 factorial end fraction plus 1 minus fraction numerator 1 over denominator 2015 factorial end fraction end cell row blank equals 1 end table$

Jadi, nilai dari $fraction numerator 1 over denominator 2015 factorial end fraction plus sum from k equals 1 to 2014 of fraction numerator k over denominator open parentheses k plus 1 close parentheses factorial end fraction$ adalah 1