Pertanyaan

Hitunglah 1 2 + 1 1 + 2 2 + 2 1 + 3 2 + 3 1 + ... + 201 6 2 + 2016 1 .

Hitunglah $\frac{1}{1 ^{2} + 1} + \frac{1}{2 ^{2} + 2} + \frac{1}{3 ^{2} + 3} + ... + \frac{1}{201 6 ^{2} + 2016}$ .

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Klaim

I. Sutiawan

Master Teacher

Mahasiswa/Alumni Universitas Pasundan

Jawaban terverifikasi

Jawaban

nilai dari adalah .

nilai dari $begin mathsize 12px style fraction numerator 1 over denominator 1 squared plus 1 end fraction plus fraction numerator 1 over denominator 2 squared plus 2 end fraction plus fraction numerator 1 over denominator 3 squared plus 3 end fraction plus... plus fraction numerator 1 over denominator 2016 squared plus 2016 end fraction end style$ adalah begin mathsize 12px style 2016 over 2017 end style

begin mathsize 12px style 2016 over 2017 end style

Pembahasan

Ingat! Sehingga: Jadi, nilai dari adalah .

Ingat!

Sehingga:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 1 over denominator 1 squared plus 1 end fraction plus fraction numerator 1 over denominator 2 squared plus 2 end fraction plus fraction numerator 1 over denominator 3 squared plus 3 end fraction plus... plus fraction numerator 1 over denominator 2016 squared plus 2016 end fraction end cell row blank equals cell fraction numerator 1 over denominator 1 left parenthesis 1 plus 1 right parenthesis end fraction plus fraction numerator 1 over denominator 2 left parenthesis 2 plus 1 right parenthesis end fraction plus fraction numerator 1 over denominator 3 left parenthesis 3 plus 1 right parenthesis end fraction plus... plus fraction numerator 1 over denominator 2016 left parenthesis 2016 plus 1 right parenthesis end fraction end cell row blank equals cell fraction numerator 1 over denominator 1 cross times 2 end fraction plus fraction numerator 1 over denominator 2 cross times 3 end fraction plus fraction numerator 1 over denominator 3 cross times 4 end fraction plus... plus fraction numerator 1 over denominator 2016 cross times 2017 end fraction end cell row blank equals cell 1 over 1 minus up diagonal strike 1 half end strike plus up diagonal strike 1 half end strike minus up diagonal strike 1 third end strike plus up diagonal strike 1 third end strike minus up diagonal strike 1 fourth end strike plus... plus up diagonal strike 1 over 2016 end strike minus 1 over 2017 end cell row blank equals cell 1 minus 1 over 2017 end cell row blank equals cell fraction numerator 2017 minus 1 over denominator 2017 end fraction end cell row blank equals cell 2016 over 2017 end cell end table end style$

Jadi, nilai dari $begin mathsize 12px style fraction numerator 1 over denominator 1 squared plus 1 end fraction plus fraction numerator 1 over denominator 2 squared plus 2 end fraction plus fraction numerator 1 over denominator 3 squared plus 3 end fraction plus... plus fraction numerator 1 over denominator 2016 squared plus 2016 end fraction end style$ adalah .