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Mirrors > Home > MPE Home > Th. List > divsqrtsumo1 | Structured version Visualization version GIF version |
Description: The sum Σ𝑛 ≤ 𝑥(1 / √𝑛) has the asymptotic expansion 2√𝑥 + 𝐿 + 𝑂(1 / √𝑥), for some 𝐿. (Contributed by Mario Carneiro, 10-May-2016.) |
Ref | Expression |
---|---|
divsqrtsum.2 | ⊢ 𝐹 = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑛)) − (2 · (√‘𝑥)))) |
divsqrsum2.1 | ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿) |
Ref | Expression |
---|---|
divsqrtsumo1 | ⊢ (𝜑 → (𝑦 ∈ ℝ+ ↦ (((𝐹‘𝑦) − 𝐿) · (√‘𝑦))) ∈ 𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpssre 12399 | . . 3 ⊢ ℝ+ ⊆ ℝ | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ℝ+ ⊆ ℝ) |
3 | divsqrtsum.2 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑛)) − (2 · (√‘𝑥)))) | |
4 | 3 | divsqrsumf 25561 | . . . . . 6 ⊢ 𝐹:ℝ+⟶ℝ |
5 | 4 | ffvelrni 6853 | . . . . 5 ⊢ (𝑦 ∈ ℝ+ → (𝐹‘𝑦) ∈ ℝ) |
6 | rpsup 13237 | . . . . . . 7 ⊢ sup(ℝ+, ℝ*, < ) = +∞ | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → sup(ℝ+, ℝ*, < ) = +∞) |
8 | 4 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ+⟶ℝ) |
9 | 8 | feqmptd 6736 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ+ ↦ (𝐹‘𝑦))) |
10 | divsqrsum2.1 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿) | |
11 | 9, 10 | eqbrtrrd 5093 | . . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ+ ↦ (𝐹‘𝑦)) ⇝𝑟 𝐿) |
12 | 5 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝐹‘𝑦) ∈ ℝ) |
13 | 7, 11, 12 | rlimrecl 14940 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
14 | resubcl 10953 | . . . . 5 ⊢ (((𝐹‘𝑦) ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((𝐹‘𝑦) − 𝐿) ∈ ℝ) | |
15 | 5, 13, 14 | syl2anr 598 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ((𝐹‘𝑦) − 𝐿) ∈ ℝ) |
16 | 15 | recnd 10672 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ((𝐹‘𝑦) − 𝐿) ∈ ℂ) |
17 | rpsqrtcl 14627 | . . . . 5 ⊢ (𝑦 ∈ ℝ+ → (√‘𝑦) ∈ ℝ+) | |
18 | 17 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (√‘𝑦) ∈ ℝ+) |
19 | 18 | rpcnd 12436 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (√‘𝑦) ∈ ℂ) |
20 | 16, 19 | mulcld 10664 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (((𝐹‘𝑦) − 𝐿) · (√‘𝑦)) ∈ ℂ) |
21 | 1red 10645 | . 2 ⊢ (𝜑 → 1 ∈ ℝ) | |
22 | 16, 19 | absmuld 14817 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (abs‘(((𝐹‘𝑦) − 𝐿) · (√‘𝑦))) = ((abs‘((𝐹‘𝑦) − 𝐿)) · (abs‘(√‘𝑦)))) |
23 | 18 | rprege0d 12441 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ((√‘𝑦) ∈ ℝ ∧ 0 ≤ (√‘𝑦))) |
24 | absid 14659 | . . . . . . 7 ⊢ (((√‘𝑦) ∈ ℝ ∧ 0 ≤ (√‘𝑦)) → (abs‘(√‘𝑦)) = (√‘𝑦)) | |
25 | 23, 24 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (abs‘(√‘𝑦)) = (√‘𝑦)) |
26 | 25 | oveq2d 7175 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ((abs‘((𝐹‘𝑦) − 𝐿)) · (abs‘(√‘𝑦))) = ((abs‘((𝐹‘𝑦) − 𝐿)) · (√‘𝑦))) |
27 | 22, 26 | eqtrd 2859 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (abs‘(((𝐹‘𝑦) − 𝐿) · (√‘𝑦))) = ((abs‘((𝐹‘𝑦) − 𝐿)) · (√‘𝑦))) |
28 | 3, 10 | divsqrtsum2 25563 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (abs‘((𝐹‘𝑦) − 𝐿)) ≤ (1 / (√‘𝑦))) |
29 | 16 | abscld 14799 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (abs‘((𝐹‘𝑦) − 𝐿)) ∈ ℝ) |
30 | 1red 10645 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 1 ∈ ℝ) | |
31 | 29, 30, 18 | lemuldivd 12483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (((abs‘((𝐹‘𝑦) − 𝐿)) · (√‘𝑦)) ≤ 1 ↔ (abs‘((𝐹‘𝑦) − 𝐿)) ≤ (1 / (√‘𝑦)))) |
32 | 28, 31 | mpbird 259 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ((abs‘((𝐹‘𝑦) − 𝐿)) · (√‘𝑦)) ≤ 1) |
33 | 27, 32 | eqbrtrd 5091 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (abs‘(((𝐹‘𝑦) − 𝐿) · (√‘𝑦))) ≤ 1) |
34 | 33 | adantrr 715 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦)) → (abs‘(((𝐹‘𝑦) − 𝐿) · (√‘𝑦))) ≤ 1) |
35 | 2, 20, 21, 21, 34 | elo1d 14896 | 1 ⊢ (𝜑 → (𝑦 ∈ ℝ+ ↦ (((𝐹‘𝑦) − 𝐿) · (√‘𝑦))) ∈ 𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 class class class wbr 5069 ↦ cmpt 5149 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 supcsup 8907 ℝcr 10539 0cc0 10540 1c1 10541 · cmul 10545 +∞cpnf 10675 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 − cmin 10873 / cdiv 11300 2c2 11695 ℝ+crp 12392 ...cfz 12895 ⌊cfl 13163 √csqrt 14595 abscabs 14596 ⇝𝑟 crli 14845 𝑂(1)co1 14846 Σcsu 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-o1 14850 df-lo1 14851 df-sum 15046 df-ef 15424 df-sin 15426 df-cos 15427 df-pi 15429 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-lp 21747 df-perf 21748 df-cn 21838 df-cnp 21839 df-haus 21926 df-cmp 21998 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-xms 22933 df-ms 22934 df-tms 22935 df-cncf 23489 df-limc 24467 df-dv 24468 df-log 25143 df-cxp 25144 |
This theorem is referenced by: (None) |
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